NLMS_September 2019

NEWSLETTER Issue: - September ETHICS IN DRINFELD MODULES COMPUTERS MATHEMATICS AND FACTORISATION AND PROOF

EDITOR-IN-CHIEF COPYRIGHT NOTICE Iain Mo att (Royal Holloway, University of London) News items and notices in the Newsletter may iain.mo [email protected] be freely used elsewhere unless otherwise stated, although attribution is requested when EDITORIAL BOARD reproducing whole articles. Contributions to June Barrow-Green (Open University) the Newsletter are made under a non-exclusive Tomasz Brzezinski (Swansea University) licence; please contact the author or photog- Lucia Di Vizio (CNRS) rapher for the rights to reproduce. The LMS Jonathan Fraser (University of St Andrews) cannot accept responsibility for the accuracy of Jelena Grbic´ (University of Southampton) information in the Newsletter. Views expressed Thomas Hudson (University of Warwick) do not necessarily represent the views or policy Stephen Huggett (University of Plymouth) of the Editorial Team or London Mathematical Adam Johansen (University of Warwick) Society. Bill Lionheart (University of Manchester) Mark McCartney (Ulster University) ISSN: 2516-3841 (Print) Kitty Meeks (University of Glasgow) ISSN: 2516-385X (Online) Vicky Neale (University of Oxford) DOI: 10.1112/NLMS Susan Oakes (London Mathematical Society) Andrew Wade (Durham University) NEWSLETTER WEBSITE The Newsletter is freely available electronically Early Career Content Editor: Vicky Neale at lms.ac.uk/publications/lms-newsletter. News Editor: Susan Oakes Reviews Editor: Mark McCartney MEMBERSHIP Joining the LMS is a straightforward process. For CORRESPONDENTS AND STAFF membership details see lms.ac.uk/membership. LMS/EMS Correspondent: David Chillingworth Policy Roundup: John Johnston SUBMISSIONS Production: Katherine Wright The Newsletter welcomes submissions of fea- Printing: Holbrooks Printers Ltd ture content, including mathematical arti- cles, career related articles, and microtheses EDITORIAL OFFICE from members and non-members. Submis- London Mathematical Society sion guidelines and LaTeX templates can be De Morgan House found at lms.ac.uk/publications/submit-to-the- 57–58 Russell Square lms-newsletter. London, WC1B 4HS Feature content should be submitted to the [email protected] editor-in-chief at iain.mo [email protected]. Charity registration number: 252660 News items should be sent to [email protected]. COVER IMAGE √ Notices of events should be prepared us- 3 ing the template at lms.ac.uk/publications/lms- A formal proof that is irrational. See the newsletter and sent to [email protected]. feature starting on page 32. (Image courtesy of For advertising rates and guidelines see Zak Buzzard.) lms.ac.uk/publications/advertise-in-the-lms- newsletter.

CONTENTS NEWS The latest from the LMS and elsewhere 4 LMS BUSINESS Reports from the LMS 13 FEATURES The Importance of Ethics in Mathematics 22 Polynomial Factorisation using Drinfeld 27 EARLY CAREER Modules 32 REVIEWS Computers and Mathematics 37 The Belgian Mathematical Society 38 40 Microthesis: Hypergraph Saturation Irregularities 45 From the bookshelf 51 OBITUARIES In memoriam 54 EVENTS Latest announcements CALENDAR All upcoming events

4 NEWS LMS NEWS LMS Elections and Donating to the LMS Annual General Meeting Making a donation to the LMS has just become easier! Voting for the LMS Elections for Council and There is now a ‘donate’ button on the main menu Nominating Committee will open on 25 Octo- bar near the top of the LMS webpage (lms.ac.uk), ber 2019. The slate of candidates can be found enabling anyone to make an online donation to the at lms.ac.uk/about/council/lms-elections and an Society. online forum for discussion is available at dis- The LMS gives grants totalling in excess of £680,000 cussions.lms.ac.uk/lmselections. per year to support many mathematical activities. In addition, members are to be asked to vote By far the greatest part of our income currently on changes to the LMS Standing Orders — the comes from publications, which one can view as a Royal Charter, Statutes, and By-Laws. The cur- way of recycling money from everyone’s work as rent Standing Orders, together with the proposed authors, editors and referees into the community. changes, can be found on the LMS website at However, as the publishing industry moves to vari- tinyurl.com/lmsstandingorders. ous forms of open access, the income we get from A vote on the proposed changes will be taken at this resource is likely signi cantly to diminish. The the Annual General Meeting, but LMS Council has threat to publications income, combined with the agreed that members who are unable to attend current unpredictability of the nancial sector, mean will be able to place an online proxy vote from 25 that this aspect of our income has become crucial. October to 25 November, through the indepen- Additional donations will help the Society maintain dent Electoral Reform Services, or if preferred and if possible increase its level of support for all of by completing a hard copy proxy voting form that its objectives. will be available from the LMS website. In the past the Society has bene tted from donations Instructions for each of the ballots i.e. a) the from many individuals. Two in particular stand out: elections to Council and Nominating Committee £1,000 from Lord Rayleigh in 1874, which supported and b) by proxy on the changes to the Standing the printing of the LMS Proceedings and rescued the Orders, will be sent to members by email or post Society from collapse, and the bequest of £50,000 before the ballots open. Members are encour- by G.H. Hardy in 1963 which was completely trans- aged to check that their contact details are up formative for the Society. to date at lms.ac.uk/user. Alongside the suggested amounts for donations, you This year the AGM will be held at Goodenough will also see a line for a ‘De Morgan Donation’ of College, Mecklenburgh Square, London WC1N 2AB £1865 or more (no prizes for working out where this on Friday 29 November at 2.30 pm. Please note notable number comes from!) This is a new venture the change of date to the end of the month and which it is hoped will encourage anyone with the the slightly earlier start time. All those attend- resources to do so to support the Society with a ing the AGM will be required to register on the substantial donation. In recognition of their support, day. The registration desk will be open from 1.30 De Morgan Donors will from time to time receive pm to allow enough time for members to verify invitations to lectures and other special events. their details and receive voting cards prior to the For those in a position to do so, you are also encour- start of the meeting. The results of the Coun- aged to think about leaving a legacy to the LMS in cil and Nominating Committee elections will be your will. Advice on how to do this can also be found announced at the meeting, as will the results of on the web page. the vote on the changes to the Standing Orders. Of course, a donation can also be added when you pay the annual membership fee. Whether your Fiona Nixon donation is large or small, the Society really does Executive Secretary value your support.

NEWS 5 Stop Press! DeMorgan@ This proportion is remarkably uniform across the 32 departments with EU postdocs. The results showed Unfortunately the big People’s Vote march in Lon- that there was a greater percentage of UK nationals don recently changed its date to October 19th which in the postdoc population in northern areas of the means that transport in and out of London is likely to UK than in southern areas. be very crowded on that day. Those planning to come Funding source: sources of funding were gathered to DeMorgan@21 are therefore advised to reserve into 8 groups. The diversity of sources was rather train seats and hotels well in advance. We apologise encouraging. As expected, EPSRC was by far the to anyone who was planning to go on the march. largest single funding source at 34% (being twice as Registration for this event has re-opened with space big as any other category); the EU, other Research for another 20 participants. These will be allocated Councils, and the postdoc’s own institution were the on a rst-come- rst-served basis. If you are inter- other main categories, at about 17% each. In several ested please register at tinyurl.com/y4r935w9. of the larger departments the amount of EU and EPSRC funding was about the same. The e ect of Survey of Mathematics Postdocs losing EU funding is likely to be signi cant. The survey also identi ed the proportion of funding to di erent The LMS has recently published the results of its 2017 Survey of Postdoctoral Researchers in the Mathemat- elds, both overall and per funding source. Generally, ical Sciences in the UK. This survey was undertaken the patterns here conformed to expectations. by the LMS Research Policy Committee in recognition Gender: the survey looked at gender balance both by of the fact that there was little overall understanding department and by eld. The departmental balance of postdoctoral activity in the UK, in terms of the showed that the overall average percentage of female size of the population, its distribution in subject area postdocs was 23%. The survey identi ed a large vari- and geographical terms, its origin and its source of ation, from 38% female in industrial mathematics funding. to 11% in PDEs and analysis, 10% in mathematical The survey was conducted in autumn 2017 with a physics and integrable systems, and 6% in number census date of 31 October 2017. Individual heads of theory. department were asked to supply information about Postdocs and the REF: As expected there is a the postdoctoral researchers in their department. strong relationship between REF performance and They were asked for data about gender, national- numbers of postdocs. It seems that there are typ- ity, home department, country of undergraduate ically very few postdocs in departments with less degree, eld of interest and source of funding, with than 20 FTE REF returns, and beyond that range a reassurances about the con dentiality of detailed ratio of 1 postdoc per 2.4 research active sta look information. The initial request was followed up with like a rather general trend across all sizes. several reminders and then collated and processed The full results of the survey can be accessed at in 2018. tinyurl.com/yyrh7fnv. We are very grateful to depart- Geography: the survey reported 756 postdoctoral ments for their cooperation in helping us to carry researchers in total — a much larger gure than was out this valuable exercise. It is intended to repeat the initially expected. A few departments did not partici- survey at regular intervals. Comments and queries pate in the survey, so the total number will in fact be are of course very welcome. higher than this gure. 25% of these were from the UK, and 43% were from the EU outside of the UK. John Greenlees LMS Vice-President CONFERENCE FACILITIES De Morgan House offers a 40% discount on room hire to all mathematical charities and 20% to all not-for-profit organisations. Call 0207 927 0800 or email roombookings@ demorganhouse.co.uk to check availability, receive a quote or arrange a visit to our venue.

6 NEWS LMS Honorary Members tureship in Applied Mathematics for his leadership in numerical linear algebra, numerical stability analysis, LMS Honorary Members Ed Witten (left) and Don Zagier and communication of mathematics. At the Society meeting on 28 June 2019, the Dr Alexandr Buryak of the University of Leeds is LMS elected Professor Edward Witten (Institute for awarded a Whitehead Prize in recognition of his out- Advanced Study, Princeton) and Professor Don Zagier standing contributions to the study of moduli of (Max Planck Institute for Mathematics) as honorary curves and integrable systems. members of the Society. Professor David Conlon of the University of Oxford Edward Witten occupies an unrivalled position is awarded a Whitehead Prize in recognition of his amongst contemporary mathematical and theoreti- many contributions to combinatorics. His particular cal physicists. He has made profound contributions expertise is Ramsey theory, where he has made fun- to the development of contemporary physics, includ- damental contributions to both the arithmetic and ing topological quantum eld theory, string theory, graph-theoretic sides of the subject. M -theory and quantum gravity. Dr Toby Cubitt of University College London is Don Zagier is an outstanding mathematician who awarded a Whitehead Prize in recognition of his out- has made major contributions in number theory, par- standing contributions to mathematical physics, in ticularly to the theory of modular forms, and in its particular the interconnections between quantum interactions with other areas of mathematics and information, computational complexity, and many- mathematical physics. body physics. Dr Anders Hansen of Cambridge University is LMS prize winners awarded a Whitehead Prize for his contributions to computational mathematics, especially his develop- The Society extends its congratulations to the fol- ment of the solvability complexity index and its cor- lowing 2019 LMS prize winners and thanks to all the responding classi cation hierarchy. nominators, referees and members of the Prizes Professor William Parnell of the University of Committee for their contributions to the Commit- Manchester is awarded a Whitehead Prize for highly tee’s work this year. novel and extensive research contributions in the A De Morgan Medal is awarded to Professor Sir Andrew Wiles FRS of the University of Oxford for elds of acoustic and elastodynamic metamaterials his seminal contributions to number theory and for and theoretical solid mechanics, as well as excellence his resolution of ‘Fermat’s Last Theorem’ in particu- in the promotion of mathematics in industry. lar, as well as for his numerous activities promoting Dr Nick Sheridan of the University of Edinburgh is mathematics in general. awarded a Whitehead Prize for his ground breaking Professor Ben Green FRS, of the University of contributions to homological mirror symmetry and Oxford, is awarded a Senior Whitehead Prize for his the structure of Fukaya categories. ground breaking results in additive combinatorics, The Berwick Prize is awarded to Dr Clark Barwick analytic number theory and group theory. of the University of Edinburgh, for his paper On the Professor Nicholas Higham FRS, of the University algebraic K -theory of higher categories, published in of Manchester, is awarded a Naylor Prize and Lec- the Journal of Topology in 2016, which proves that Waldhausen’s algebraic K -theory is the universal homology theory for ∞-categories, and uses this uni- versality to reprove the major fundamental theorems of the subject in this new context. Dr Eva-Maria Graefe of Imperial College London is awarded an Anne Bennett Prize in recognition of her outstanding research in quantum theory and the inspirational role she has played among female stu- dents and early career researchers in mathematics and physics.

NEWS 7 LMS prize winners Sir Andrew Wiles Ben Green Nick Higham De Morgan Medal Senior Whitehead Prize Naylor Prize Alexandr Buryak David Conlon Anders Hansen Whitehead Prize Whitehead Prize Whitehead Prize Nick Sheridan William Parnell Toby Cubitt Whitehead Prize Whitehead Prize Whitehead Prize Clark Barwick Eva-Maria Graefe Berwick Prize Anne Bennett Prize

8 NEWS Ken Brown awarded the David Beyond this long-term involvement with the LMS, Ken Crighton Medal has made numerous further contributions to the UK Mathematical Sciences, including as a member of the The LMS and IMA have Research Assessment Exercise (RAE) Mathematics awarded the 2019 David subpanel in 1996, as Vice-Chair of this panel in 2001, Crighton Medal to Ken and then as Chair of the RAE Pure Mathematics Sub- Brown, Professor of panel in 2008. He was also a member of the Research Mathematics at the Uni- Excellence Framework (REF) expert advisory group versity of Glasgow, for in 2008-09. He has been a member of the EPSRC his seminal contribu- College since 1996 and served on the EPSRC Mathe- tions to noncommuta- matical Sciences Strategic Advisory Team (SAT) rst tive algebra and for his as a member from 2013 to 2015 and subsequently remarkable record of service and dedication to the as its Chair from 2015 to 2017. UK mathematics community. Ken has the rare ability to apply subtle ring-theoretic Read the full citation at tinyurl.com/y5ubvpg3. ideas to solve important problems in related areas. In the 1970s he solved the zero-divisor question for Calderon Prize abelian-by- nite groups, introducing the key homo- logical techniques which would form the basis of all LMS member Carola Schönlieb (University of Cam- later major progress in this area; according to For- bridge) has been awarded the 2019 Calderon Prize for manek, he made “the most important and original her work in image processing and partial di erential contribution to the problem since Higman’s [1940] equations. The Inverse Problems International Asso- work.” In the 1980s he introduced the class of homo- ciation awards the Calderon Prize to a researcher logically homogeneous rings: in the last ve years, under the age of 40 who has made distinguished these have been key in the study of noncommutative contributions to the eld of inverse problems broadly geometry, derived categories, and moduli spaces. His de ned. Carola is the rst female mathematician to focus then shifted to the theory of quantum groups receive this award. She won an LMS Whitehead prize and Hopf algebra where he harnessed the combi- in 2016 and is the current leader of European Women nation of Hopf algebras and homological algebra to in Mathematics. con rm important conjectures of Kac-Weisfeiler and DeConcini-Kac-Procesi in representation theory. In Forthcoming LMS Events the last decade he has proved core results in sev- eral topics: Noetherian Hopf algebras; number the- The following events will take place in the next three ory through Iwasawa algebras; Poisson geometry in months: Lie theory; and symplectic re ection algebras. Ken wishes to acknowledge his many collaborators in his Prospects in Mathematics Meeting: 6–7 Septem- research. He was elected a Fellow of the Royal Society ber, Lancaster (tinyurl.com/y4c9aaxk). of Edinburgh in 1993. Midlands Regional Meeting: 11 September, Notting- Ken has mirrored his distinguished international ham (tinyurl.com/y5vtaytx). mathematical career with extraordinary service to Popular Lectures: 19 September, University of Birm- the UK Mathematical Sciences community. He sat on ingham (tinyurl.com/hu58wjk). the London Mathematical Society Council for almost DeMorgan@21: 19 October, London. two decades, including terms as Vice-President from Joint Meeting with the IMA: 21 November, Reading 1997-99 and 2009-17. During this time he was instru- (tinyurl.com/y4sdm74b). mental in the development of the voice of the Coun- Computer Science Colloquium: 13 November, Lon- cil for the Mathematical Sciences, providing critical don (tinyurl.com/cscoll19). input to consultations and leading a variety of task LMS/BCS-FACS Evening Seminar: 21 November, forces, particularly helping highlight the important London (tinyurl.com/yyc9oyse). issues that a ect the Mathematical Sciences people Graduate Student Meeting: 29 November, London pipeline. Beyond this long-term involvement with the (tinyurl.com/yy58t78v). LMS, Ken has made numerous further contributions Society Meeting and AGM: 29 November, London to the UK Mathematical Sciences. (tinyurl.com/yy58t78v). A full listing of upcoming LMS events can be found on page 54.

NEWS 9 OTHER NEWS The rst excerpt is a table from page 240 which provides a schema for succinctly representing Turing Alan Turing honoured on new £ machines. The table gives a complete description banknote of how to specify such machines and therefore can be thought of as one of the rst examples of a pro- gramming language. Alan Turing banknote concept The second excerpt, from page 241, is a sequence of Turing machine transitions that helps explain how An announcement made by the Bank of England on to encode a Turing machine as a number. The more 15 July 2019 has con rmed that Alan Turing will be modern analogue of what Turing describes is how the character to feature on the reverse side of the to take an abstract representation of a computer new £50 banknote, which will come into circulation program and convert it into a binary sequence of in 2021. 0s and 1s so that it can be stored on a disc or in Alan Mathison Turing OBE FRS (1912–1954) was a math- the memory of a computer. The idea that a program ematician, computer scientist, logician, cryptanalyst, can be stored as a number, and used as data (by an philosopher and theoretical biologist who was instru- operating system) in order to execute the program, mental in formalising the concepts of algorithm and is hugely important. computation. Turing worked as a code-breaker dur- ing the second world war and is widely accredited Turing went on further in his article to describe large with having helped bring an earlier end to the war. classes of real numbers whose binary expansions The story of his life has had wide implications for are computable by his machines; to describe a ‘uni- changes in political, legal and social attitudes towards versal machine’ that could serve the purpose of an human diversity and homosexuality. operating system; and to describe the theoretical In his article ‘On computable numbers, with an limits of his machines. Ultimately, Turing showed that application to the Entscheidungsproblem’ (submit- there can be no algorithmic method for determining ted 28 May 1936, published in Proceedings of the whether or not a given mathematical statement can London Mathematical Society 42 (1937) 230–265; be proved in a certain axiomatic system. This proved tinyurl.com/y3vlud4j), Turing presented a rst model that David Hilbert’s famous Entscheidungsproblem for a general-purpose computer, later to become has no solution (which was also proved independently known as a ‘Turing machine’. by Alonzo Church). The London Mathematical Society welcomes this Alan Turing is the second mathematician to appear great exposure and boost to the public appreciation on a Bank of England banknote. A £1 banknote in of mathematics, and is delighted to have contributed circulation between 1978 and 1988 depicted Sir Isaac to the design of the banknote by giving approval and Newton. Famous Britons have featured on the reverse permission for two mathematical excerpts from this of Bank of England banknotes since 1970. Turing article to be included on the new banknote. Paul Shafer, University of Leeds Ola Törnkvist, LMS Editorial Manager

10 NEWS MATHEMATICS POLICY DIGEST Brexit threat to research New Director General, Industrial and innovation Strategy and Innovation Royal Society President, Sir Venki Ramakrishnan Jo Shanmugalingam became Director General, Indus- wrote a letter in July warning of the impact a trial Strategy, Science and Innovation at the Depart- no-deal Brexit will have on research and innova- ment for Business, Energy and Industrial Strategy tion. More information and the letter are available at (BEIS) on 15 July 2019. More information about the new tinyurl.com/yyas8lz3. Director General is available at tinyurl.com/y2hch6cn. Exploring the workplace for LGBT+ Digest prepared by Dr John Johnston physical scientists Society Communications O cer The Institute of Physics, the Royal Society of Chem- Note: items included in the Mathematics Policy Digest istry and the Royal Astronomical Society, have con- are not necessarily endorsed by the Editorial Board or ducted a comprehensive survey to gather data the LMS. from across the community — giving new insights into the current workplace environment for LGBT+ physical scientists. The full report is available at tinyurl.com/y38x9kxh. EUROPEAN MATHEMATICAL SOCIETY NEWS From the EMS President Funding for Mathematical Research The last months have been full of activities. Here Volker Mehrmann (President of the EMS) writes: As I would like to mention a few. A highlight was the President of the European Mathematical Society I award of the Abel Prize to Karen Uhlenbeck, the rst would like to point out a very urgent and unfavourable female recipient, on 21 May. This was followed by a situation for the funding of mathematics in Europe. day of talks related to her work. These can be viewed The European Research Council (ERC) budget for at tinyurl.com/yx8p4v6f. each discipline is allocated each year in proportion to The restructuring of the EMS publishing house is the number of proposals and the requested budget underway. The new organization, in the form of a received. It has been observed that, since the found- limited company owned by the EMS, was established ing of the ERC, the budget for mathematics in the at the end of March. The hiring process for the three funding streams (advanced, consolidator, and new management is in progress. The new publishing starting grants) has dropped to almost half, because house will immediately face a major challenge, with there are not enough applications. the increasing importance of open access publishing, There may be several reasons for this decline in in particular Plan S of the European Commission. The applications, e.g. low acceptance rate, the feeling EMS has reacted to this plan, pointing the possible that certain sub elds of mathematics have small consequences for small publishers and the mathe- chances, or the fact that for interdisciplinary research matics community. See tinyurl.com/y54uw828. of mathematics with other sciences it is di cult to Another major concern is research funding for math- get funding. Also in mathematics there are often ematics in Europe, which is constantly decreasing complaints that the maximal possible budgets are even within the European Research Council. See the too large. next item, appealing to EMS members to become All this is partially right, but not submitting applica- more active on all levels. tions leads to a vicious cycle, and further decline of

NEWS 11 mathematics funding. How can we counteract this identified the top three science priorities in order for unfortunate development? First of all, there is no Japan to lead the fourth industrial revolution and to reason to apply for the full possible budget if this even go beyond its limits: mathematics, mathematics, is not appropriate for a research project, smaller and mathematics!” proposals are very welcome, and second we mathe- maticians should be more self-con dent in writing Bernoulli Center (CIB) proposals. It is not a wasted time, even if one is not funded. In several European countries there is even The CIB in Lausanne has announced the following sched- ule of research programmes: Dynamics with Structures nancial support for proposals that make it to the (1 July – 31 December 2019); Functional Data Analysis (1 second round but do not get funded due to budget January – 30 June 2020); Locally Compact Groups Acting restrictions. on Discrete Structures (1 July – 31 December 2020); Dy- It is very important that applications are encour- namics, Transfer Operators, and Spectra (1 January – 30 aged throughout the mathematical community and June 2021). For more details see http://cib.epfl.ch. the EMS is planning to create an initiative to sup- port applicants. So please distribute this information European Congress of Mathematics within your community. The preparations for the 8th European Congress of Volker Mehrmann Mathematics in Portorož, Slovenia, from 5 to 11 July President of the EMS 2020 are proceeding energetically. A full list of plenary and invited speakers at for the 8th European Congress Academia–industry roundtable of Mathematics in Portorož, Slovenia, is available at 8ecm.si/program. The EMS Applied Mathematics Committee notes with in- terest the minutes of an academia-industry roundtable EMS News prepared by David Chillingworth organised by the ministry of economy in Japan (March LMS/EMS Correspondent 2019) (tinyurl.com/y3o54aud) which claims “we have OPPORTUNITIES EMS Prizes Events within Europe and speakers with strong rela- tions to European academic institutions will be given Calls for nominations of candidates for ten European preference. Closing date for applications is 30 Septem- Mathematical Society Prizes as well as The Felix Klein ber 2019. For further details and nomination forms go Prize and The Otto Neugebauer Prize for the History to the EMS website and click on Scientific Activities. of Mathematics are still open: details can be found at the European Mathematical Society website. Maryam Mirzakhani Prize in Mathematics: call for nominations EMS Call for Proposals In recognition of Dr Mirzakhani’s remarkable life and The European Mathematical Society (EMS) Meetings achievements, the National Academy of Sciences Committee is calling for nominations or proposals for has established a newly named Maryam Mirzakhani speakers and scientific events in 2020. The EMS is Prize in Mathematics (formerly the NAS Award in willing to provide support to cover the cost of EMS Mathematics, established in 1988 by the American Lecturers and Distinguished Speakers, and to give par- Mathematical Society in honour of its centennial). tial support to the organization of EMS Weekends and The prize will be awarded biennially for exceptional of EMS Summer Schools of high scientific quality and contributions to the mathematical sciences by a mid- relevance. The EMS is committed to increasing the par- career mathematician. Nominations for the inaugural ticipation of women in research in mathematics and Mirzakhani Prize are due by 7 October 2019. For more its applications. Efforts to give opportunities to math- information see tinyurl.com/yykobras. ematicians of both genders and to integrating early career mathematicians will be particularly appreciated.

12 NEWS Women in Mathematics and other working in all areas of mathematics and mathematics Diversity Events education, including teachers, researchers and post- graduate students who can demonstrate a bene t A call for expressions of interest in running Women in from attending ICME-14. Mathematics Days, Girls in Mathematics Days and a Download the application form at tinyurl.com/yyagyo23. new event for 2019/20, Diversity in Mathematics Day, Email the completed application form to Professor is now open. The deadline for submitting expressions Chris Budd, University of Bath ([email protected]). of interest is Friday 20 September 2019. The deadline is 30 November 2019. More details about Women in Mathematics Days are available at tinyurl.com/y4wy8px3, Girls in Mathe- Clay Research Fellowships: matics Days at tinyurl.com/y44t4mdt, and Diversity call for nominations in Mathematics Day at tinyurl.com/y2a3pmbg. The Clay Mathematics Institute calls for nomina- ICME- Bursaries tions for its competition for the 2020 Clay Research Fellowships. Fellows are selected for their research The Joint Mathematical Council of the UK has achievements and their potential to become lead- launched the ICME bursaries scheme, which is partly ers in research mathematics. All are recent PhDs, funded by the LMS. The scheme will provide nine and most are selected as they complete their thesis bursaries of £500 each to attend the 14th Interna- work. Terms range from one to ve years, with most tional Congress on Mathematical Education to be given in the upper range of this interval. Fellows are held from 12 to 19 July 2020 in Shanghai, China employed by the Clay Mathematics Institute, which (tinyurl.com/yyxbvxth). is a US charitable foundation, but may hold their The bursaries can fund travel, subsistence, child- fellowships anywhere in the USA, Europe, or else- care, registration fee or preparation of a presentation. where in the world. Nominations should be received They cannot fund salary or related costs. Applica- by 16 November 2019. To nominate a candidate see tions are encouraged from the full range of those tinyurl.com/y6stxs6o. VISITS Visit of Grigory Belousov torics and pattern-avoiding permutations. While in the UK he will visit and give talks in Newcastle, Heriot Dr Grigory Belousov (Plekhanov Russian Univer- Watt and St Andrews Universities. For further details sity of Economics, Bauman Moscow State Tech- contact [email protected]. Supported by an nical University) will visit the UK from 1 to 17 LMS Scheme 2 grant. November 2019. During his visit, he will give lec- tures at the University of Liverpool, Loughborough Visit of Albert Visser University and the University of Edinburgh. His research revolves around Singular del Pezzo Surfaces. Professor Albert Visser (Universiteit Utrecht) will For further details contact Nivedita Viswanathan visit the University of Cambridge from 7 to 29 ([email protected]). Supported by an October 2019. He is professor emeritus of logic LMS Scheme 2 grant. and a member of the Royal Netherlands Academy of Arts and Sciences; he is particularly well-known Visit of Murray Elder for his research contributions on weak systems of arithmetic. In addition to seminar presentations Dr Murray Elder (University of Technology Sydney, in Cambridge, talks will be given at the Universi- Australia) will visit the UK from 4 November to 4 ties of Oxford (14 October) and Leeds (16 Octo- December 2019. His research interests include geo- ber). For further information contact Benedikt Löwe metric group theory, complexity theory, automata ([email protected]). Supported by an LMS and formal language theory, enumerative combina- Scheme 2 grant.

LMS BUSINESS 13 LMS Council Diary — were disadvantaged in comparison to those funded A Personal View elsewhere. Council agreed that this item should be deferred for further discussion at the October 2019 Council met at De Morgan House on Friday, 28 June, a meeting. little earlier than usual, due to the LMS General Meet- We also heard reports from the IT Resources Commit- ing being held that same afternoon. As usual, the tee and from the Publication Secretary. Furthermore, meeting began with an update on the President’s ac- we received the Third Quarterly Financial Review. Fol- tivities since the last Council meeting, which included lowing a report from the Prizes Committee Council attendance at the LMS Reps Day and attendance agreed that Professor Kenneth Brown be con rmed at several of the Society’s committees. She also as the Crighton Medal recipient in 2019. reported that three members of the LMS received We then decamped to the nearby Mary Ward House honours in the Queen’s birthday list: Kenneth Brown for the General Meeting and Aitken Lecture 2019. received a CBE, Peter Ransom an MBE, and Peter Don- nelly received a Knighthood. Vice President Hobbs Brita Nucinkis reported from the European Mathematical Society Presidents’ meeting, which was held in March 2019 Perigal Artefacts in Berlin. She reported that diversity took up some part of the discussion. With the calls for EMS prizes The London Mathematical Society has been by given now published, the LMS was encouraged to promote by Daniel Miskow some items which once belonged the nominations process, particularly to address the to Henry Perigal, an LMS member famous for his need for diversity of nominations. Council agreed proof of Pythagoras’s Theorem by dissection. The that the free EMS membership for PhD students items include a collection of index cards containing should be advertised. diagrams from Euclid’s Elements, and a beautifully- The Education Secretary then reported that a work- decorated envelope made by folding paper. The enve- ing group had been set up in December 2018 to lope contains six cardboard triangles and ve quadri- address the issue of the shortage of quali ed mathe- laterals, as well as three pieces of blueish paper, matics teachers. It was felt that the group should be carefully cut and folded. The members of the Library formalised as a sub-committee of Education Com- Committee have been unable to deduce their pur- mittee, which Council approved. Council also agreed poses and would welcome elucidation from mem- to extend the Teachers CPD to allow universities to bers who are welcome to contact the LMS Librarian propose events for HE teaching and learning events. ([email protected]). The General Secretary reported that the latest ver- sion of the document containing the proposed Mark McCartney changes to the Standing Orders was now available LMS Librarian to view on the website, having received informal approval from the Privy Council. The Chair of the Women in Mathematics Committee reported on the foreword to the National Benchmark- ing Survey Report, which is the LMS’s sole contri- bution to the report; this foreword was approved. Council also agreed on the LMS Statement on Diver- sity in Mathematics. Council noted with pleasure that the Irish Mathemat- ical Society had agreed to a Reciprocity Agreement with the LMS. There was some discussion regarding the amount given to individual LMS Undergraduate Bursaries — some Council members felt that it would be detri- mental to the Society’s reputation, and the diver- sity of applicants, if recipients of LMS bursaries

14 LMS BUSINESS LMS Committee Membership Access to LMS Journals The Society o ers free online access to the Bulletin, The detailed business of the LMS is run by about 23 Journal and Proceedings of the London Mathematical committees and working groups, each usually con- Society and to Nonlinearity for personal use only. If sisting of about 10 people. Altogether this comes to you would like to receive free electronic access to a large number of people, to whom the Society is these journals, please indicate your choices either extremely grateful for this vital work. on your online membership record under the ‘Journal It is Council’s responsibility to make the appoint- Subscription’ tab or on the LMS subscription form. ments to all these committees and to turn their The relevant publisher will then contact members membership over regularly, so that (a) the broadest with further details about their subscription. possible spectrum of our membership is represented, Subscribing to the EMS and JEMS via the LMS and (b) the committees remain fresh and energetic. Members also have the option to pay their European Of course, when forming a committee, account has Mathematical Society subscription via the LMS and to be taken of many things, such as maintaining sub- subscribe to the Journal of the EMS. If you would ject and demographic balance, which means that on like to subscribe to the EMS and JEMS via the LMS, a given occasion otherwise very strong candidates indicate either on your online membership record may not always be able to be appointed. under the ‘Journal Subscription’ tab or on the LMS So we are always looking for new people! See subscription form. lms.ac.uk/about/committees for a list of committees. Payment of membership fees for EWM If you are interested, or would like to recom- LMS members who are also members of European mend a colleague, please contact James Taylor at Women in Mathematics may pay for their EWM fees [email protected] in order that Council can when renewing their LMS membership. You decide maintain a good list of potential members of its var- your category of fees: high, normal, low. Indicate your ious committees. It is not necessary to specify a category of fee either on your online membership particular committee. If you would like to know what record under the ‘Journal Subscription’ tab or on is involved, you could in the rst instance ask your the LMS subscription form. To join EWM, register LMS Departmental Representative. at tinyurl.com/y9 pl73. It is not possible to join the EWM through the LMS. Stephen Huggett Online renewal and payment General Secretary Members can log in to their LMS user account (lms.ac.uk/user) and make changes to their contact Annual LMS Subscription - details and journal subscriptions under the ‘My LMS Membership’ tab. Members can also renew their sub- Members are reminded that their annual subscrip- scription by completing the subscription form and tion, including payment for additional subscriptions, including a cheque either in GBP or USD. We do not for the period November 2019 – October 2020 is due accept payment by cheques in Euros. on 1 November 2019 and payment should be received LMS member bene ts by 1 December 2019. Payments received after this An LMS annual subscription includes the following date may result in a delay in journal subscriptions bene ts: voting in the LMS elections, free online being renewed. access to selected journals, the bi-monthly Newslet- ter, use of the Verblunsky Members’ Room at De LMS membership subscription rates 2019–20 Morgan House in Russell Square, London and use of the Society’s Library at UCL. For a full list of mem- Ordinary membership £88.00 US$176.00 ber bene ts, see lms.ac.uk/membership/member- Reciprocity £44.00 US$88.00 bene ts. Career break or part-time £22.00 US$44.00 working Elizabeth Fisher Associate membership £22.00 US$44.00 Membership & Grants Manager Associate (undergrad) £11.00 US$22.00 membership

LMS BUSINESS 15 LMS Grant Schemes holder. For those mathematicians going to their col- laborator’s institution, grants of up to £2,000 are For full details of these grant schemes, and for infor- available to support a visit for collaborative research mation on how to submit an application form, visit by the grant holder to a country in which mathemat- www.lms.ac.uk/content/research-grants. ics could be considered to be in a disadvantaged MATHEMATICS RESEARCH GRANTS position. The deadline is 15 September 2019 for applications Research Workshop Grants: Grants of between for the following grants, to be considered by the £3,000 - £5,000 are available to provide support for Research Grants Committee at its October meeting. Research Workshops held in the United Kingdom, the Conferences Grants (Scheme 1): Grants of up to Isle of Man and the Channel Islands. £7,000 are available to provide partial support for AMMSI African Mathematics Millennium Science conferences held in the United Kingdom. Awards Initiative: Grants of up to £2,000 are available to are made to support the travel, accommodation, support the attendance of postgraduate students subsistence and caring costs for principal speakers, at conferences in Africa organised or supported by UK-based research students and participants from AMMSI. Application forms for LMS-AMMSI grants are Scheme 5 eligible countries. available at ammsi.africa. Visits to the UK (Scheme 2): Grants of up to £1,500 MATHEMATICS/COMPUTER SCIENCE RESEARCH are available to provide partial support for a visitor GRANTS to the UK, who will give lectures in at least three Computer Science Small Grants (Scheme 7): separate institutions. Awards are made to the host Grants of up to £1,000 are available to support visits towards the travel, accommodation and subsistence for collaborative research at the interface of Math- costs of the visitor. It is expected the host institu- ematics and Computer Science, either by the grant tions will contribute to the costs of the visitor. holder to another institution within the UK or abroad, Joint Research Groups in the UK (Scheme 3): or by a named mathematician from within the UK Grants of up to £4,000 are available to support joint or abroad to the home base of the grant holder. research meetings held by mathematicians who have Deadline: 15 October. a common research interest and who wish to engage GRANTS FOR EARLY CAREER RESEARCHERS in collaborative activities, working in at least three The deadline is 15 October 2019 for applications for di erent locations (of which at least two must be the following grants, to be considered by the Early in the UK). Potential applicants should note that the Career Research Committee in November. grant award covers two years, and it is expected that Postgraduate Research Conferences (Scheme 8): a maximum of four meetings (or an equivalent level Grants of up to £4,000 are available to provide partial of activity) will be held per academic year. support for conferences held in the United Kingdom, Research in Pairs (Scheme 4): For those mathemati- which are organised by and are for postgraduate cians inviting a collaborator to the UK, grants of up research students. The grant award will be used to to £1,200 are available to support a visit for collabo- cover the costs of participants. rative research either by the grant holder to another Celebrating New Appointments (Scheme 9): institution abroad, or by a named mathematician Grants of up to £600 are available to provide partial from abroad to the home base of the grant holder. support for meetings held in the United Kingdom For those mathematicians collaborating with another to celebrate the new appointment of a lecturer at UK-based mathematician, grants of up to £600 are a UK university. available to support a visit for collaborative research. Travel Grants for Early Career Researchers: Collaborations with Developing Countries Grants of up to £500 are available to provide par- (Scheme 5): For those mathematicians inviting a tial travel and/or accommodation support for UK- collaborator to the UK, grants of up to £3,000 are based Early Career Researchers to attend confer- available to support a visit for collaborative research, ences or undertake research visits either in the by a named mathematician from a country in which UK or overseas. mathematics could be considered to be in a disad- vantaged position, to the home base of the grant

16 LMS BUSINESS REPORTS OF THE LMS Report: LMS Invited Lectures Thursday 23 May, organised on behalf of the Applied Probability Section of the Royal Statistical Society. Professor Søren Asmussen Professor Asmussen presented applications of phase- The LMS Invited Lectures 2019 were delivered by type distributions in life insurance, building from the Professor Søren Asmussen (Aarhus University) on graduate-level material he discussed earlier in the 20-24 May at the International Centre for Mathemat- week to the state of the art in the area. Additional, ical Sciences in Edinburgh, on advanced topics in complementary presentations were given by Burak life insurance mathematics. This course was based Buke (University of Edinburgh), Ayalvadi Ganesh (Uni- on parts of Professor Asmussen’s forthcoming book versity of Bristol) and Ronnie Loe en (University of Risk and Insurance: A Graduate Text, co-authored with Manchester) on various aspects of the theory of Mogens Ste ensen (University of Copenhagen). stochastic processes, with applications to nancial, Professor Asmussen gave a fascinating account of biological and other systems. inhomogeneous Markov models, semi-Markov mod- A relaxed schedule and the hospitable environment els, and their applications in life insurance. Topics o ered by the ICMS allowed for plenty of interac- covered in detail included unit-linked insurance, inter- tion between participants at various career stages est rate and mortality rate modelling, and dividends throughout the meeting, and also time to enjoy the and bonuses. The course consisted of nine lectures city of Edinburgh in one of the better weeks of and two tutorials, and was complemented by an weather this summer has given us. opening lecture by Takis Konstantopoulos (University of Liverpool) and a closing lecture by Andrew Cairns Fraser Daly and Sergey Foss (Heriot-Watt University). Heriot-Watt University Participants (particularly early-career researchers) were given the opportunity to present some of Report: LMS Popular Lectures their own work. During the week we heard inter- esting presentations from Jason Anquandah (Uni- Peter Higgins versity of Leeds) on Optimal Stopping in a Simple On 26 June 2019 the Institute of Education, London Model of Unemployment Insurance, Abdul-Lateef Haji- hosted the 2019 LMS Popular Lectures. The audience Ali (Heriot-Watt University) on Multilevel Monte Carlo enjoyed an evening of mathematics with a journey for E cient Risk Estimation, and Lewis Ramsden (Uni- through a life in mathematics and a look at the com- versity of Hertfordshire) on The Time to Ruin for a plexity of the human brain and some of the basic Dependent Delayed Capital Injection Risk Model. principles at work in shaping our brains. In addition, a workshop on Stochastic Processes in Peter Higgins (University of Essex) presented the rst Finance and Beyond was held on the afternoon of lecture and took the audience through his personal

LMS BUSINESS 17 mathematical journey. Going back to his roots and Report: LMS Undergraduate recounting personal episodes that led to him becom- Summer Schools ing a professional mathematician. He went on to tell the story of how he invented circular Sudoku. Hig- The LMS Undergraduate Summer Schools are annual gins recognised from his work that the knowledge he two-week courses, which are held every summer at a had accumulated could be applied to mathematical UK university. The aim is to introduce modern mathe- problems in this case relating to Burrows–Wheeler matics to the best UK undergraduates who will enter transform, which is an algorithm that rearranges char- their nal year the following autumn, and encourage acter strings into runs of similar characters and is them to think about an academic career in mathe- important in data compression. matics. The LMS Undergraduate Summer School con- sists of a combination of short lecture courses with Alain Goriely problem-solving sessions and colloquium-style talks The second lecture was presented by Alain Goriely from leading mathematicians. 50 places are available (University of Oxford). The outer surface of the brain to students per year. Gwyn Bellamy, organiser of the (cerebral cortex) is very convoluted (wrinkled) so that 2018 LMS Undergraduate Summer School at Glasgow, a maximum amount of gray matter (brain neurons) reports on the LMS Undergraduate Summer School can fit inside the skull, but how do these convolutions 2018. The 2019 LMS Undergraduate Summer School emerge and how is the brain’s geometry related to took place at the University of Leeds from 14–26 July function? Professor Goriely began by looking at the and the next LMS Undergraduate Summer School relative sizes of the brains of some famous scientists will take place at the University of Swansea from and mathematicians, with some interesting results. 12-24 July 2020. Nominations for places will open in Einstein in particular had a small brain compared with October 2019. Babbage, Helmholtz, Gauss and Dirichlet. He went on In July 2018, 53 extremely eager students from across to describe comparisons in the convolutions of Gauss the United Kingdom descended on Glasgow for a and De Morgan and discussed how the brain develops jam-packed two weeks of mathematical activities its shape by using mechanics. Gauss had complex which is the LMS Undergraduate Summer School. convolutions and De Morgan had convolutions that, Each mathematics department in the UK elects up to although ’voluminous’, were less intricate. Professor three applicants to attend, and from these we select Goriely explained how different parts of the brain are approximately 50 participants. Almost the entire cost connected to each other in brain networks and how of the two weeks is covered by the LMS. neurodegenerative diseases develop. In essence he In the rst week, lecture courses were given by Pro- showed how some of the basic principles at work in fessor Mike Prest (University of Manchester) on The shaping the brain can be explained using geometry, Compactness Theorem, by Dr Robert Gray (University scaling laws and network topology. of East Anglia) on The Word Problem in Combinato- If you weren’t able to attend the London lectures, rial Group and Semigroup Theory and by Professor there is another chance to attend the Lecture in Birm- Shaun Stevens (University of East Anglia) on Local– ingham on Thursday 19 September. You can register Global Principles in Number Theory. In the second online at lms.ac.uk/events/popular-lectures. week, lecture courses were given by Dr Martina Lanini (Università di Roma Tor Vergata) on Introduction to John Johnston Schubert Calculus, by Dr Derek Harland (University of LMS Communications O cer Leeds) on Fun with Solitons and by Dr David Bourne (Heriot-Watt University) on Optimal Transport Theory. As one would expect, the lectures were uniformly fascinating and engaging, and a real pleasure to lis- ten to. It was clear that the lecturers had spent a great deal of time and energy in preparing their lec- tures. Moreover, each lecturer had prepared a series of exercises to go with the lectures. Together with PhD and MSc students from the University of Glas- gow, the lecturers guided the students through the exercises during the exercise sessions.

18 LMS BUSINESS In addition to the lecture series, there were collo- Surprisingly, around 10 of the students managed to quium talks later in the afternoon on most days. solve all the problems. These were: There is a real hunger amongst undergraduate mathe- matics students in the UK to be exposed to research • Arithmetic and Geometry of Conway Rivers by Pro- level mathematics. The LMS Undergraduate Sum- fessor Alexander Veselov mer School is one of the few ways in which we can e ectively meet this need. Though it is, as one • Chasing the Dragon: Tidal Bores in the UK and Else- can imagine, a great deal of work to organize an where; Quantum and Hawking Radiation Analogies LMS Undergraduate Summer School, it is also an by Sir Michael Berry extremely rewarding experience and an e ective way to make a signi cant contribution to undergraduate • Faster than Fourier (pre)revisited: Vorticulture, Noise, mathematics in the UK. For those LMS members in Fractals, Escape. . . by Sir Michael Berry a position to apply to organize a summer school, I would strongly encourage you to do so. • Unexpected Connections. . . by Dr Tom Leinster I sincerely hope that the summer school will have • A Tour of the Mandelbrot Set by Dr Holly Krieger had a positive in uence on each student’s long-term • The Birch and Swinnerton-Dyer Conjecture: From the relationship with the mathematical sciences, and that we’ll see many of them studying for a PhD in the Ancient Greeks to a 1 Million Dollar Problem of the near future. I would also like to give my thanks to the 21st Century by Dr Alex Bartel lecturers and colloquium speakers for all their hard • Mathematics, Magic and the Electric Guitar by Dr work in preparing their talks, and their enthusiasm David Acheson in delivery. • Random Games with Finite Groups by Professor Colva Roney-Dougal Gwyn Bellamy University of Glasgow As one can guess from the above list, these wonder- ful talks took the students on a journey through a Report: LMS Northern Regional huge swath of exciting mathematics, all of which was Meeting and Workshop beautifully illustrated (and in one case accompanied by live music!) Conference participants An LMS Northern Regional Meeting took place in On the Sunday between the two weeks of lectures, the afternoon of Tuesday 28 May 2019, followed by there was an organised bus trip to Balmaha on the a wine reception and dinner, and a Workshop on shores of Loch Lomond. All the students braved the Higher Dimensional Homological Algebra was held on weather to climb the nearby Conic Hill, where there Wednesday 29 May 2019 at Newcastle University. were spectacular views of the loch and nearby Ben Lomond; at least this was the case until about half way up, the summit itself was unfortunately shrouded in mist for the day. Afterwards we retired to the pub at the base of the hill for a well-deserved drink. As organiser, by far the most enjoyable aspect of the summer school was interacting with the partic- ipants. It was energizing being surrounded by so many bright, enthusiastic and hardworking young people! They really gave it their all, and I think this ensured that they got the most out of the two weeks. It was particularly gratifying to see them very quickly bond, seeming to be old friends by the end of the rst week. It was also a great experience to be in exercises classes where the students were asking questions throughout and racing to try and solve the problems. To provide motivation for the students to work on mathematics questions (as it turned out, this was not required at all!), a series of mathematical problems were set, with prizes for the best solutions.

LMS BUSINESS 19 The LMS Northern Regional Meeting and the work- The workshop was organised by Ilke Canakci and shop brought to the UK international experts in the Peter Jorgensen. The organisers thank the LMS representation theory of nite dimensional algebras and the School of Mathematics, Statistics and to present their work on the most recent develop- Physics at Newcastle University for nancial sup- ments and the speakers included many of the most port. The website of the workshop can be found at renowned experts in the area. The workshop also tinyurl.com/y4qc2gwt. gave early career mathematicians such as PhD stu- dents and postdocs an opportunity to present their Ilke Canakci work. In total there were nine speakers, of whom one Newcastle University was a PhD student and three were postdocs. Report: LMS Graduate Student There were two general talks at the Northern Meeting and the General Meeting Regional Meeting on the latest developments in the of the Society & Aitken Lecture eld. These talks were given by Karin Baur (University Aitken Lecturer Bakh Khoussainov of Graz and University of Leeds) on CM Modules for The LMS Graduate Student Meeting and the General Grassmannians and by Sibylle Schroll (University of Meeting of the Society & Aitken Lecture were held Leicester) on Geometric Models and Derived Invariants on Friday 28 June 2019, a warm and sunny summer’s for Gentle Algebras. day, at Mary Ward House in London. The morning session was kicked o by Dr Robert D. Gray, of the The workshop was focused on Higher Dimensional University of East Anglia, whose lecture served as Homological Algebra. This area concerns d -cluster an introduction to the material which would be pre- tilting subcategories, which are structures found at sented in lectures later in the day. Ideas surrounding the cusp of algebra and combinatorics. They have a rich algebraic structure, investigated by homological nite presentations of algebraic structures, recur- methods, and there are many examples relating to sive enumerability, and notions such as the word higher dimensional structures in combinatorics, such problem were all introduced. This lecture was fol- as cyclic polytopes. lowed by a number of graduate student presenta- tions, where the broad selection of topics ranged There were seven talks at the workshop, two of from climate models, modular representation the- which were expository talks by Ste en Oppermann ory, and polyhedral combinatorics, among others. By (Norwegian University of Science and Technology) popular vote, two graduate speakers were then each on (d + 2)-angulated cluster categories and by Hugh awarded a prize: Aras Asaad, for a presentation on Thomas (Universite du Quebec a Montreal) on Tropical using homology to detect computer generated faces, Coe cient Dynamics for Higher-Dimensional Cluster and Carl-Fredrik Nyberg Brodda, for a presentation Categories. Talks focusing on recent developments on decision problems for groups and semigroups. of higher homological algebra were by Francesca After the morning’s Graduate Student Meeting, the Fedele (Newcastle University) on A (d + 2)-angulated afternoon’s General Meeting was opened by Profes- generalization of a theorem by Bruning, by Martin Her- sor Caroline Series. While the morning meeting had schend (Uppsala University) on Wide subcategories of d -Cluster tilting subcategories for higher Auslander algebras, by Karin Marie Jacobsen (NTNU) on d-abelian Quotients of d + 2-angulated Categories, by Gustavo Jasso (Universitat Bonn) on Generalised BGP Re ection Functors via Recollements and by Sondre Kvamme (Université Paris-Sud) on d -abelian Categories are d - cluster Tilting. The atmosphere during the meeting was very pleas- ant and convivial. The frequent co ee breaks, the wine reception and dinner followed by the Northern Regional Meeting and the lunch at the workshop pro- vided great opportunities for discussions, informal exchanges of ideas and for establishing professional contacts. There were 31 participants, 10 of whom were PhD students.

20 LMS BUSINESS seen a respectable audience, the room was quickly Paul Shafer lled up by new attendees at the beginning of the While many new de nitions were planted in the minds of the audience members during the course of the General Meeting. Two new Honorary Members of the presentation, there was no lack of prepared ground Society were appointed, and the announcement of in which to place them owing to the earlier lectures; the 2019 LMS Prize Winners followed. Following tra- the end result was an excellent insight into this fas- dition, several new members of the Society were cinating eld of research laying at the intersection then given the opportunity to sign their names in the of many di erent areas. Members’ Book. To serve as an introduction to the Finally, to nish o the day, a wine reception fol- later Aitken Lecture, Dr Paul Shafer, of the University lowed by a conference dinner were both held at the of Leeds, presented a thorough lecture on the topic Ambassadors Bloomsbury Hotel, rounding o a most of computability, which served well to extend the informative and enjoyable day in June. material presented earlier in the day. As could be expected on a warm day in June, the Carl-Fredrik Nyberg Brodda room — which at this point had become lled to the University of East Anglia brim with mathematicians — had grown hot, to the point that a collective sigh of relief could be heard when the windows were nally opened. Refreshed by this ood of fresh air, the audience members all seemed sharpened and ready to enjoy the nal event of the General Meeting; this was the Aitken Lecture, given by Professor Bakh Khoussainov of the University of Auckland. A central idea therein, for which the audience was now more than prepared as a result of the earlier lectures, was the question of in which circumstances one can (or, indeed, cannot) nd nitely presentable expansions of a given algebra. This question, and the context in which it is asked, touches on a number of di erent topics, including immune algebras, residual niteness of algebras, and the generalised Burnside problem for groups. Records of Proceedings at LMS meetings Northern Regional Meeting at Newcastle University: 28 May 2019 The meeting was held at the Herschel Building, Newcastle University, as part of the workshop on Higher Dimensional Homological Algebra. Over 30 members and guests were present for all or part of the meeting. The meeting began at 3.10 pm, with the Vice-President, Professor Cathy Hobbs, in the Chair. There were 31 members elected to Membership at this Society Meeting. Four members signed the Members’ Book and were admitted to the Society by the Vice-President during the meeting. There were no Records of Proceedings signed at this meeting of the Society. Dr Ilke Canakci introduced the rst lecture, given by Dr Karin Baur (ETH Zurich/Leeds) on CM-Modules for Grassmannians. Dr Ilke Canakci then introduced the second lecture, given by Dr Sibylle Schroll (Leicester) on Geometric Models and Derived Invariants for Gentle Algebras. The Vice-President thanked the speakers for their talks, and further extended her warm thanks to the local organisers, Dr Ilke Canakci and Professor Peter Jorgensen, for hosting such a well-structured and interesting meeting, and additionally thanked the attendees for coming to the Northern Regional Meeting. Dr Ilke Canakci thanked the speakers, and invited all those present to attend a wine reception held in the Penthouse of the Herschel Building. A Society Dinner was held following the wine reception in the Herschel Building.

LMS BUSINESS 21 Records of Proceedings at LMS meetings General Meeting: 28 June 2019 The meeting was held at Mary Ward House, Tavistock Square, London. Over 60 members and visitors were present for all or part of the meeting. The meeting began at 3.30 pm with the President, Professor Caroline Series, FRS, in the Chair. On a recommendation from Council, it was agreed to elect Professor Charles Goldie and Professor Chris Lance as scrutineers in the forthcoming Council elections. The President invited members to vote, by a show of hands, to ratify Council’s recommendation. The recommendation was rati ed unanimously. The President, on Council’s behalf, proposed that following people be elected to Honorary Membership of the Society: Professor Ed Witten, of the Institute for Advanced Study at Princeton University and Professor Don Zagier, of the University of Bonn. This was approved by acclaim. The President read a short version of the citations, which would be published in full in the Bulletin of the London Mathematical Society. The President then introduced the General Secretary who gave a report on the review of the Charter, Statutes and By-Laws. The President then announced the awards of the prizes for 2019: De Morgan Medal: Professor Sir Andrew Wiles (University of Oxford); Senior Whitehead Prize: Professor Ben Green (University of Oxford); Naylor Prize & Lectureship in Applied Mathematics: Professor Nicholas Higham (University of Manchester); Whitehead Prizes: Dr Alexandr Buryak (University of Leeds), Professor David Conlon (University of Oxford), Dr Toby Cubitt (University College London), Dr Anders Hansen (University of Cambridge), Professor William Parnell (University of Manchester), and Dr Nick Sheridan (University of Edinburgh); Berwick Prize: Dr Clark Barwick (University of Edinburgh); Anne Bennett Prize: Dr Eva-Maria Graefe (Imperial College London). Fourteen people were elected to Ordinary Membership: Dr Alla Detinko, Professor Johny Doctor, Mr Dalebe Gnandi, Dr Marina Iliopoulou, Dr David Kimsey, Dr Tomasz Lukowski, Dr Jesu Martinez Garcia, Professor James Maynard, Mr Luthais McCash, Professor Monica Musso, Dr Louis Theran, Dr Sofya Titarenko, Mr Bunonyo Wilcox and Dr Gordon Woo. Twelve people were elected to Associate Membership: Mr Isarinade Ayodeji Felix, Mr Eduard Campillo- Funollet, Mr Carl Dawson, Mr Fabio Ferri. Dr Maciej Matuszewski, Mr Dimitris Michailidis, Dr Isaiah Odero, Mr Wasim Rehman, Mr Mark Scott, Dr Joni Ter äväinen, Mr Nicholas Williams and Dr Mehdi Yazdi. Three people were elected to Reciprocity Membership: Mr Rizwan Kassamally, Dr Kwara Nantomah and Dr Alan Sola. Three members signed the book and were admitted to the Society. The President announced the dates of the next two Society Meetings to be held on 11 September in Nottingham as part of the Midlands Regional Meeting and on 21 November in Reading as part of the Joint Meeting with the Institute of Mathematics and its Applications. The President also reminded members that the date of the Annual General Meeting had moved from 15 November to 29 November. The President announced that, to celebrate the 21st Anniversary of the Society’s move into De Morgan House, there would an event on 19 October 2019. The President further announced that the Society was seeking to recruit a new Editor in Chief for the LMS Newsletter. The President introduced a lecture given by Dr Paul Shafer (University of Leeds) on An Introduction to Computable Functions and Computable Structures. Following a break for tea, the President introduced the Aitken Lecture by Professor Bakh Khoussainov (University of Auckland) on Semigroups, Groups, Algebras, and their Finitely Presented Expansions. At the end of the meeting, the President thanked both speakers for their brilliant lectures. The President also thanked Robert Gray (University of East Anglia), who gave the main talk at the Graduate Student Meeting in the morning. Aras Assad (University of Buckingham) and Carl-Fredrik Nyberg Brodda (University of East Anglia) were also congratulated on winning the prizes for the best Graduate Student talks. After the meeting, a reception was held at the Ambassador Bloomsbury Hotel in the Enterprise Suite, followed by a dinner at the Number Twelve restaurant in the Ambassador Bloomsbury Hotel.

22 FEATURES The Importance of Ethics in Mathematics MAURICE CHIODO AND TOBY CLIFTON Mathematics is useful because we can nd things to do with it. With this utility comes ethical issues relating to how mathematics impacts the world. Now, more than ever, we mathematicians need to be aware of these, as our mathematics, and our students, are changing society. In the rst of a two-part series on Ethics in Mathematics, we address why, as mathematicians, we need to consider the ethics of what we do. Mathematics and the world beauty and intrinsic interest, rather than its applica- tions in science and industry. It’s as though we are We study one of the most abstract areas of human studying a form of abstract art; far from real world knowledge: mathematics, the pursuit of absolute impact or considerations, and only fully appreciated truth. It has unquestionable authority. But, in some by a small number. Despite all this, government and sense, absolute truths have absolutely no meaning. industry pay for our work; one suspects they don’t The statement “2 + 3 = 5” is an absolute truth, just do this for the sake of our intellectual stimula- but what does it mean? Its meaning and utility tion. If our work is completely abstract and detached are added later when people who understand the — an art form, so to speak — then shouldn’t we seek statement reconcile it with the physical world. It is funding from those who fund abstract art? So what the mathematically-trained who interpret and apply might be the value that science councils and industry mathematics to the real world and thereby assign it see in what we do? It is not only the mathematical meaning; through this it becomes useful. results that we produce, but also the mathemati- Indeed, it is clear that mathematics is one of the cians we train. Our mathematics makes a di erence, most useful and re ned tools ever developed. When and our students go out and do real things with something is useful, however, it can often also be their training. If it is the case that our work is being harmful; this can be either through deliberate misuse funded because it has perceived impact, then surely or ignorance. The humble knife provides an illus- we should query and understand why we are being tration of the principle; in order to use such a tool paid to do it. responsibly, one must be made aware — often by There is already much discussion of ethics in the those who rst introduce you to it — of the poten- mathematical community. However, these discus- tial dangers. If a primitive tool like a knife can be sions usually focus exclusively on issues within the so useful and harmful at the same time, then what community. These are important and many of us are is mathematics capable of? Mathematics has many already familiar with them: from improving diversity more applications, and by the same reasoning must and inclusivity, to widening participation in mathe- also have a greater potential to do ill. As mathemati- matics, to addressing instances of plagiarism and cians we are seldom warned of this. Other disciplines publishing irregularities. These are pressing concerns. such as law, medicine, and engineering have, for a Every discipline engages with such intrinsic ethical long time, addressed the potential for harm within issues. These, however, are not our focus in this their eld. We, as the practitioners and wielders of article. Mathematics is one of the few disciplines that mathematics, need to be similarly aware, and adjust fails to address extrinsic ethical issues; those con- our actions accordingly. Otherwise we can, and some- cerning how the community impacts wider society. times do, cause harm with our work. But how could This particularly includes ethical implications relat- mathematics possibly be harmful, and what exactly ing to the applications of mathematics and work of might this harm be? mathematicians. It is these extrinsic ethical issues In this article our emphasis is on the experiences of that we are trying to raise awareness of. Our concern pure mathematicians, although our arguments apply is not so much that mathematicians are deliberately equally to applied mathematicians, statisticians and malign, but instead that they fail to recognise these computer scientists. Many of us (although certainly extrinsic ethical issues. Indeed, most of the mathe- not all) are motivated to study mathematics by its maticians we have come across would baulk at the thought of acting unethically; the problem, instead, is

FEATURES 23 that many do not recognise that mathematical work as such it’s now possible to specify who doesn’t see can have such an e ect. an advert. This allows advertising campaigns that are selective, that contain adverts that contradict Some case studies each other, and that are impossible to externally Having recognised that mathematics is useful scrutinise. In short, adverts can now be used to ma- because it can be applied, and that with these appli- nipulate individual people. This becomes particularly cations come extrinsic ethical issues, we now con- dangerous when applied to political advertising. Us- sider two concrete examples: the global nancial ing large data sets obtained through social media, crisis (GFC) of 2007–8 and targeted advertising. it is possible to pro le the political persuasions and The GFC was one of the de ning events that shaped preferences of an individual. Machine learning has the modern global economy. Its repercussions have become the main tool of the trade here, and it is been felt around the world, with many su ering the mathematically trained doing it [3]. These ad- a decline in living standards. The causes of the verts can even deceive by appearing non-partisan. GFC are complex, however, there is consensus that For instance, one can send an advert saying “Voting mathematical work played a vital role. An impor- is important; make sure you vote” only to those who tant factor is thought to have been the misuse of might be inclined to vote for your party. Whatever Collateralised Debt Obligations (CDOs). These saw the strategy, these types of adverts are increasingly mathematicians pool large collections of interest- prevalent, and it is thought that such tactics in u- bearing assets (mostly mortgages), then ‘cut the pool enced the 2016 US election and UK referendum on into pieces’ to form a collection of interest-bearing EU membership. It is we mathematicians who make products. Mathematically these products had less all this possible. Cambridge Analytica, one of the overall risk and thus higher value than the original organisations alleged to have been involved in such assets. They were traded wildly. The mathematics advertising, had a small team of no more than 100 behind their construction is highly non-trivial, requir- data scientists [4], some of whom were trained math- ing stochastic calculus, di erential equations, etc. ematicians. Regardless of one’s political persuasion, Research mathematicians, beginning with the work it is clear that this sort of work is deceptive and of Black and Scholes, and later Li, derived a model dangerous, and that mathematicians are enabling it. and pricing formula for CDOs. Though it took a deep Ultimately, it is mathematicians who make up part understanding of mathematics to derive these mod- of the teams specifying how such targeting works els, only a more super cial understanding (at the and carrying it out. undergraduate level) was required to apply and to trade them. As a result, their users may not have The impact of mathematicians fully appreciated their limitations or inner workings. As a result of the pace and scale at which mod- Mathematics — which by itself is sure and certain ern technology operates, through use of internet — seemed to explain their value, and so most were connectivity and readily-available fast computation, happy. Unfortunately, some of the assumptions did the consequences of the actions of mathematicians not hold. For example, the model assumed there are more quickly realised and far-reaching than ever wasn’t tail dependence in the default risk of under- before. A mathematician in a big tech company can lying assets, but there was; for instance when two modify an algorithm, and then have it deployed mortgaged houses were on the same street. In the almost immediately over a user base of possibly bil- end, the risk was not properly accounted for, and lions of people. Even on a smaller scale, we have when house prices declined it led to the write-down seen that a small number of mathematicians, despite of $700billion of CDO value from 2007 to 2008. The limited resources, can have a vast impact on the rest is history. world; targeted advertising exempli es this. Our next example is targeted advertising. Adverts If you model a physical system, such as gravity, then have always been placed so as to catch the eye of your model is falsi able. If the model does not accu- their desired audience. However, now that people rately re ect the physical system, then on application possess portable internet-connected devices and it clearly fails — your rocket doesn’t launch prop- social media accounts, it has become possible to tar- erly. You know when a model was good because the get adverts at the individual level. Nowadays, these rocket makes it to the moon and back. Modelling a can be tuned to t very speci c demographics, and nancial system is more di cult, as the system is

24 FEATURES a ected by the application of the model. A pricing point that most people will not be interested, it must algorithm, if widely-used to buy or sell a product, be remembered that some people will be su ciently in uences the market for the product in question. desperate for credit that there will always be some How does a model model its own impact? takers. These companies scrape an applicant’s social So now what happens if you are modelling the future media looking for actions they perceive to re ect behaviour of people by predicting something like creditworthiness. These could include places the per- ‘How likely is a particular individual charged with son visits, the hours they sleep, the ‘quality’ of the a crime to reo end with a serious o ence, a non- friends they have, and so on. This approach is unfalsi- serious o ence, or not reo end, in the next 24 months?’ Furthermore, what if that is being used to able, lacks proper regulation, and has the potential determine what prosecution and sentencing mecha- to harm society since the extension of credit is a nisms are applied to that person?1 If you predict that mechanism of social mobility. If such a process, one a person will reo end seriously in 24 months, and that is enabled by mathematically-trained people, they don’t (after being released or acquitted), then starts having negative impact, who is accountable? you might observe that. But what if they are found Ultimately, we must live in the world that we and our guilty and sentenced to 25 months, with the choice students create, and we must ponder whether there of judicial process based on your prediction? How do is a sense in which we are partially responsible. you test whether your prediction was correct? Now we have a serious ethical issue: we are using math- Do these ethical issues arise in academia? ematical reasoning to make decisions about people But what about mathematicians working in academia; that impact their lives, and in many of these cases are any of these ethical issues relevant to them? Con- we can never know whether the decisions made were sider a pure mathematician, a number theorist, say. desirable or appropriate. Is it right to use mathemat- Suppose they develop an algorithm for fast factorisa- ics in such a way without careful re ection? tion. Should they publish it? If so, when, where, and We now face an ethical dilemma. Do we limit our- how? If not, what should they do? Should they have selves to falsi able claims, or do we allow ourselves thought about it beforehand? We have asked many to make claims, make decisions, and initiate actions mathematicians this exact question, and a typical that are unfalsi able? We are of course entitled to do response is “I would publish it on arXiv immediately. the latter, however we should then bear in mind that It’s my right to publish whatever mathematical work we have lost mathematical certainty. Furthermore, if I do.” (Not all mathematicians give such a response, we do this, we should broaden our perspective and but many do.) When pressed on the consequences training so that we can incorporate as many aspects of publishing such an algorithm in that way — for of society as possible. instance the breaking of RSA encryption in a chaotic manner and the ensuing collapse of internet com- Concerns for the future merce and the global economy that would follow — So what is on the horizon for mathematicians? Is it one explained “Well, it’s their fault for using RSA. su cient to simply look at the above list of cases and It’s not my problem.” Of course, responsible disclo- avoid those speci c actions or industries entirely? sure is a complicated topic, and one that is heavily Unfortunately not; new mathematics produces new debated by security researchers. But with an exam- ethical issues every day. Such a future example may ple like this, ethics has crept into the world of the lie in alternative credit scoring. This is starting to be pure mathematics researcher in academia. If an area done by new companies who lack access to standard as abstract as number theory is not ‘safe’ from ethi- datasets that established credit-scoring agencies cal considerations, is there any mathematical work have (such as nancial records, bill payment history, that is? Can a pure mathematician hide from ethical etc.). They instead use di erent datasets such as issues in academia? What about a statistician, or an social media pro les, in some cases requesting full applied mathematician? Or do ethical questions arise access to social media accounts by asking for login for all mathematicians regardless of where we do our credentials [2]. While this sounds undesirable to the mathematical work? 1The Harm Assessment Risk Tool (HART) developed by the Durham Constabulary, which uses random forest machine learning, is used to make such predictions, and then determine if an accused criminal is to be o ered the opportunity of going through the Checkpoint program (tinyurl.com/y4vxrd77) which is an alternative to criminal prosecution aimed at reducing re-o ending.

FEATURES 25 Why management can’t guide us might”. True, someone else might, but they may not Some mathematicians (academic or industrial) may be easy to nd, or even exist at all. Now the power think that, since they are not directly involved in the of meaningful objection has returned to the mathe- application of their work, they need not consider its matician. Whether you choose to take the pragmatic extrinsic ethical implications. After all, we just do the perspective that there are not many mathematicians, maths, and so it’s ‘not our problem’. This oft-held or the logical perspective of the contrapositive, your belief is generally associated with the perception that objection means something. Some mathematicians there exist people and structures above us (man- take this idea even further, and make a conscious agers, supervisors, advisory boards, etc.) who will decision to take a seat at the table of power, e ect- intervene to prevent us from doing anything that we ing positive change from the managerial level. This ought not to. We work on the abstract problems, they happens in various areas: in academia, in industry, worry about why. But can we rely on management to and even in politics. This is discussed in more detail do this e ectively? Will they vet our work, to ensure in [1] as ‘the third level of ethical engagement’. that its use is aligned with the values of society? At each stage of separation from mathematical work, Why the law can’t guide us some understanding of it is lost. It is di cult for The problem extends beyond management. We may a manager to understand all of the mathematical think that the law provides a clear description of work we do, and its limitations when applied and what is and is not acceptable to society, and thereby used. It is the nature of management that managers presumably what is and is not ethical. However, this will only have partial knowledge of the work being misses the point for several reasons. Firstly, the law done. There would be no point in a manager repro- is not an axiomatised system; it is interpreted by ducing the work of all of the people under them, courts rather than by machines. This is a type of sys- and mathematics is such that if you don’t ‘do it for tem with whose details mathematicians are generally yourself’ then there is a chance you may not fully not familiar. Furthermore, there is the problem that understand it. Given this fact, there is always an onus the law will always lag behind technological develop- on the individual mathematician to consider the eth- ment; we cannot expect lawmakers to have done our ical implications of what is being done. Of course, it mathematics before we do it ourselves. Additionally, must also be considered that managers might have the processes by which laws are made are (deliber- other values, perhaps more aligned with the objec- ately) slow, requiring public consultation, votes, and tives of the organisation than of wider society. We implementation periods. Consider the case of the should understand and anticipate this. General Data Protection Regulation (GDPR). It started Some managers may go so far as to try to manipu- to be written in 2011, only came in to e ect in 2018, late us. For instance, if we voice objection at what and is thought by many to be already out of date. we have been asked to do, they may try to quash it Finally, it can be the case that lawmakers lack a full with the classic argument: “if you don’t do it, then understanding of the ne details of the subject at someone else will”. At a rst glance, this seems hand. For example, a member of the UK Science convincing, however, it fails on two counts when and Technology Select Committee, Stephen Metcalfe, referring to mathematical work. First, there are not declared at a public outreach event that “one solu- that many mathematicians in the world. We possess tion to algorithmic bias is the use of algorithms to a unique set of skills and abilities, and it requires check algorithms, and the use of algorithms to check years of training to produce a good mathematician, training data”. Ultimately, the law is not there to serve even when starting with someone who has the right as moral advice; there are plenty of immoral things interests and re exes. Given the scarcity of mathe- one can do that do not break any laws. As such, it is maticians, this argument fails in practice. Moreover, not well-suited as a source of ethical advice. as mathematicians we understand its contrapositive; Thus, if we can’t rely on management and we can’t the original statement is equivalent to “If no-one else rely on lawmakers and regulators, then who can we does it, then you will”. This is, of course, absurd. The rely on? The answer is as obvious as it is di cult argument has as the implicit underlying assumption to admit: ourselves. The only way mathematicians that the task being requested will de nitely be com- can try to prevent their work from being used to do pleted. If no-one else builds me a nuclear bomb, then harm is if they think about it themselves. No one will you? What we should really be considering here is else can, so we must. the argument “If you don’t do it, then someone else

26 FEATURES A growing awareness of Ethics in Mathematics of the Newsletter, we’ll further explore why such teaching has not yet occurred in mathematics, and Awareness that mathematicians need to consider ex- outline how one might go about giving such directed trinsic ethical issues is building in the community. In teaching of Ethics in Mathematics. 2018 the head of mathematics at Oxford, Professor Mike Giles, commented at a panel discussion event: Acknowledgements “Cambridge Analytica is interesting from one point of view in that, if you’d asked me 20 years ago whether We wish to thank Piers Bursill-Hall and Dennis Müller mathematicians at the PhD level needed to be ex- for their valuable discussions over the past few years, posed to ideas of ethics, I would have said ‘Clearly, and Dennis in particular for his comments and sug- that is irrelevant to mathematicians’. Now I really gestions on previous versions of this article. think that this is something we have to think about. In the same way that engineers have courses looking FURTHER READING at ‘What it means to be a professional engineer’, and [1] M. Chiodo, P. Bursill-Hall, Four levels of ethi- ‘Ethics, and your responsibilities as an engineer’, I cal engagement, Ethics in Mathematics Discussion think that is something that we have to think about Papers, 2018/1 (2018). as mathematicians now.” Moreover, arxiv.org is cur- [2] Systems and methods for using online social rently revising the description of their mathematics footprint for a ecting lending performance and tag History and Overview to include “Ethics in Mathe- credit scoring, Patent No. US 8,694.401 B2 , 8 April matics” as a sub-category. 2014. As part of their formal training, few mathematicians [3] Cambridge Analytica: how did it turn clicks into have ever been told about extrinsic ethics before. votes?, theguardian.com, tinyurl.com/yy89e5vl, Previous generations of mathematicians have evaded May 6th 2018. this crucial point, and in the process have possibly let [4] Secrets of Silicon Valley, Series 1 Episode 2: society down. It rests on the current and upcoming The Persuasion Machine, see recording at 30:17. generations to pick up this idea, before it’s too late. BBC documentary, 2017 Mathematicians always take a generation or more to accept a new and fundamental idea about the nature Maurice Chiodo of their subject; debates about the admissibility of zero as a number provide such an example. We’re Maurice is a postdoc- at a similar juncture again. Now some say “Surely toral research fellow there’s no use in considering ethical issues in mathe- at the University of matics”, but by the time our students are professors Cambridge, and lead and industry leaders, they may well be saying “Of investigator of the Cam- course we should be considering ethical issues in bridge University Ethics mathematics!” But why hasn’t the mathematical com- in Mathematics Project munity taken this on board already? Why wasn’t this (ethics.maths.cam.ac.uk). He’s developing a teaching done 100 years ago, by the likes of Gödel and Russell? programme and curriculum to teach mathematicians Two reasons come to mind. Firstly, the dangers were about the ethical implications of their work. less proximate, since much of today’s technology simply did not exist. Secondly, every mathematics Toby Clifton undergraduate was already exposed to philosophy, as it formed part of every university education. Thus, Toby is a recent grad- exposure to Ethics in Mathematics, in its own right, uate of astrophysics, was less urgently needed. and is the current pres- So if Ethics in Mathematics has become so important ident of the Cambridge to mathematicians, then how might we teach it to University Ethics in them in a relevant and useful way, without foisting Mathematics Society an entire philosophy degree upon them? Disciplines (cueims.soc.srcf.net). such as law, medicine, and engineering have long taught their undergraduate students about extrinsic ethics in their respective elds. In the next issue

FEATURES 27 Polynomial Factorisation Using Drinfeld Modules ANAND KUMAR NARAYANAN The arithmetic of Drinfeld modules has recently yielded novel algorithms for factoring polynomials over nite elds; a computational problem with applications to digital communication, cryptography and complexity theory. We o er a gentle invitation to these developments, assuming no prior knowledge of Drinfeld modules. Factoring polynomials modulo a prime uct of irreducible polynomials of the same degree. By Fermat’s little theorem, x . To Let Fq denote the nite eld of integers modulo an extract the product of linear q −x = of af(∈xFq),xta−kea the odd prime number q . Polynomials Fq [x] over Fq share factors striking analogies with integers, yet we begin with greatest common divisor gcd(xq − x, f(x)). To extract an algorithmic distinction. While factoring integers products of degree two factors, degree three factors remains a notoriously di cult problem, factoring poly- and so on iteratively, look to the succinct expression nomials in Fq [x] is long known to be easy, at least with access to randomness. xqd − x = p(x ) p:deg(p)|d Polynomial factorisation over nite elds is not a for the product of monic irreducible polynomials p(x) mere curiosity, but has many applications. In num- of degree dividing d . At the d th iteration, with smaller ber theory, nite elds arise as residue elds of degree factors already removed, gcd(xqd − x, f(x)) global elds such as number elds. While deter- yields the product of degree d irreducible factors. mining the splitting of a prime in a number eld, one factors a polynomial de ning the number eld Care in handling xqd is required for its degree is expo- modulo the prime. Several instances of polynomial nential in n log q . All we need is xqd mod f(x), easily factorisation appear while factoring integers using accomplished by a sequence of q th powers modulo quadratic/number eld sieve algorithms or while per- f(x), each performed by repeated squaring. Better forming index calculus to compute discrete loga- still, xqd mod f(x) can be rapidly computed with a rithms, both foundational problems in analysing the fast algorithm to compose two polynomials modulo security of cryptographic systems. In digital com- f(x) in concert with q th powers. Kaltofen and Shoup munication, polynomial factorisation aids the con- devised an ingenious improvement over this naive struction of certain error correcting codes (BCH and iteration resulting, in a signi cant speed up. The cyclic redundancy codes), structures vital to reliable Kaltofen–Shoup algorithm implemented using the transmission of information in the presence of noise. modular composition algorithm of Kedlaya–Umans performs distinct degree factorisation with run time Let us recount a simple polynomial factorisation exponent 3/2 in the degree n and 2 in log q . algorithm. We are given a monic f(x) ∈ Fq [x] of degree n whose factorisation into irreducible poly- Equal degree factorisation nomials f(x) = thatim=i1sptih(xe)piis(xs)oaurgehdt.isAtisnscut.mTehifs(xis) Distinct degree factorization leaves us with the prob- is square free, lem of factoring polynomials all of whose irreducible without loss of generality for there are algorithms to factors are of the same known degree d . All known rapidly reduce to this )sapnedciawlecasseee.kItaltgaokreitshmn lsotgh2aqt algorithms for this task with polynomial runtime in bits to write down f(x log q are randomized. Even for the simplest case run in time polynomial in n and log q . Berlekamp was of factoring a quadratic polynomial into two linear the rst to show there is a randomized polynomial factors, no unconditional deterministic polynomial time algorithm, but we follow a di erent two step time algorithms are known. It is closely related to the process. problem of nding a quadratic nonresidue modulo a given large prime. Distinct degree factorisation The following randomized algorithm can be traced The rst step, known as distinct degree factorisation, backed to ideas of Gauss and Legendre. For a uni- decomposes f(x) into factors each of which is a prod-

28 FEATURES formly random a(x) of degree less than n, Finite Drinfeld modules gcd q d −1 −1, f(x ) Drinfeld introduced the modules bearing his name as an analogue of elliptic curve complex multiplication a(x) 2 theory. He in fact called them elliptic modules. Drin- feld modules and their generalisations have played gives a random factorisation of degree d irre- a crucial role in the class eld theory of function ducible factors. This follows since raising a(x) to the (q d − 1)/2-th power modulo a degree d irreducible elds and in proving the global Langlands conjecture polynomial results in either 1 or −1, depending on over function elds for G Ln. We settle for a con- whether a(x) reduces to a quadratic residue or not. crete simple notion of Drinfeld modules su cient Remarkably, this computation can be performed with for our context. Throw the q th power Frobenius σ run time exponent 1 in the degree n using an algo- into Fq [x] resulting in Fq (x) σ , the skew polynomial rithm of von zur Gathen and Shoup implemented ring with the commutation rule σ u(x) = u (x)q σ, for with the aforementioned Kedlaya–Umans modular all u(x) ∈ Fq [x]. A rank-2 Drinfeld module over Fq (x) composition. is (the Fq [x] module structure on the additive group In summary, the best known polynomial factorisa- scheme over Fq (x) given by) a ring homomorphism tion algorithms have run time exponent 3/2 in the degree with the bottleneck being distinct degree fac- φ : Fq [x] −→ Fq (x) σ torisation. To lower this exponent is an outstanding x −→ x + gφ(x) σ + ∆φ(x) σ2 problem. In fact, to lower this exponent, it su ces for there to be an algorithm that merely estimates for some gφ(x) ∈ Fq [x] and nonzero ∆φ(x) ∈ Fq [x]. the degree of some irreducible factor. To better understand the map, it is instructive to compute by hand to where x2, x3 and so on, get Drinfeld modules and polynomial factorisation mapped. By design, b(x) maps to a polynomial in σ The use of Drinfeld modules to factor polynomials with constant term b(x), over nite elds originated with Panchishkin and Potemine [3]. Drawing inspiration from Lenstra’s 2 deg(b) elliptic curve integer factorization, they recast the role of the group of rational points on random elliptic b(x) −→ φb := b(x) + φb,i (x)σi . curves modulo primes with random nite Drinfeld modules. i =1 We describe three Drinfeld module based algorithms for polynomial factorisation. The rst two were Consider an Fq [x] algebra M . In our algorithms to fac- devised in [2] and the third in [1]. The rst esti- tor f(x), M will often turn out to be M = Fq [x]/(f(x)). mates factor degrees using Euler–Poincaré charac- One way to make an Fq [x] algebra M into an Fq [x] teristics in hopes of speeding up distinct degree module is to retain the addition and scalar multiplica- factorisation. The second is a Drinfeld analogue of tion but simply forget the multiplication. The Drinfeld Lenstra’s algorithm, closely related to the aforemen- module φ endows a new Fq [x] module structure to M tioned algorithm of Panchishkin and Potemine [3]. by twisting the scalar multiplication. For b(x) ∈ Fq [x] Our exposition begins with a short account of - and a ∈ M , de ne the scalar multiplication nite Drinfeld modules followed by Euler–Poincaré the characteristic and Frobenius distributions, important 2 deg(b) ingredients in the rst two algorithms. The third algorithm involves Drinfeld modules with complex b(x) a := φb(a) = b(x)a + φb,i (x)aqi , multiplication with an analogue of Deligne’s congru- ence playing a vital role. It is also the fastest of the i =1 three algorithms, with runtime complexity matching the best known algorithms, in theory and practice. where the arithmetic on the right is performed in The hope is, the rich arithmetic of Drinfeld modules the Fq [x] algebra M . Let φ(M ) denote the new Fq [x] will inform new algorithms to beat the 3/2 exponent module structure thus endowed to M . barrier. Euler–Poincaré characterisitic Cardinality is an integer valued measure of the size of a nite abelian group (equivalently, a nite Z-module). A convoluted de nition is to assign as the cardinality of a cyclic group of prime order the corresponding prime, and for cardinality of nite abelian groups that sit in an exact sequence to be multiplicative. The

FEATURES 29 Euler–Poincaré characteristic χ is an Fq [x]-valued our polynomial to factor? The multiplicativity of the cardinality measure of a nite Fq [x] module de ned Euler–Poincaré characteristic implies completely analogously. For a nite Fq [x] module A, χ(A) ∈ Fq [x] is the monic polynomial such that: χ φ Fq [x] (f(x)) = χ φ Fq [x] (pi (x)) • If A Fq [x]/(p(x)) for a monic irreducible p(x), i then χ(A) = p(x). • If 0 → A1 → A → A2 → 0 is exact, then χ(A) = = pi (x) + tφ,pi (x) = f(x) + tφ,f(x), χ(A1) χ(A2). i For the Fq [x] module φ(Fq [x]/(f(x))) featuring in our algorithms, the Euler–Poincaré characteristic for some tφ,f(x) of degree at most sf /2, where sf χ(φ(Fq [x]/(f(x)))) has a simple linear algebraic inter- denotes the degree of the smallest degree factor of pretation: the characteristic polynomial of the linear f(x). Thus, we have an extension of Gekeler’s bound map φx on Fq [x]/(f(x)). In particular, it is a degree n to reducible polynomials polynomial that can be computed e ciently. χ φ Fq [x] (f(x)) = f(x) + tφ,f(x), Frobenius distribution of Drinfeld modules ≤sf /2 Let us put our newly de ned Fq [x] modules and car- dinality measure χ to use. Take an elliptic curve E implying f(x) and χ(φ(Fq [x]/(f(x)))) agree at the high over the rational numbers and reduce it at a prime p. degree coe cients! The number of agreements tells The Fp -rational points E(Fp ) famously form a nite us information about the smallest factor degree. abelian group with cardinality p + 1 up to an error For a randomly chosen φ, tφ,f(x) likely has degree determined by the Frobenius trace tE,p . The Hasse exactly sf /2 (with probability at least 1/4). The bound, considered the Riemann hypothesis for 2el√lipp-. number of agreements does not merely bound but tic curves over nite elds, asserts that |tE,p |≤ determines the degree of the smallest factor. To claim this probability, one needs to prove for a ran- Thereby, domly chosen φ, the Frobenius traces corresponding to the irreducibles of smallest degree do not conspire #(E(Z/(p))) = p + 1 − tE,p . yielding cancellations. To this end, we seek equidistri- bution formulae for the Frobenius traces. Analogous −2√p ≤ ≤2√p to elliptic curves, there is a correspondence between the number of isomorphism classes of Drinfeld mod- Gekeler established the following Drinfeld module ules with a given trace and Gauss class numbers in analogue. Take a Drinfeld module φ, a monic irre- certain imaginary quadratic orders. The latter can be ducible polynomial p(x), and consider the resulting computed using analytic class number formulae. Fq [x] module φ(Fq [x]/(p(x))). Its Euler–Poincaré char- An algorithm to estimate the degree of the smallest acteristic equals degree factor of a given f(x) is now apparent. Pick a Drinfeld module φ (by choosing gφ, ∆φ at random χ φ Fq [x] (p(x)) = p(x) + tφ,p(x) , of degree less than n). Compute the Euler–Poincaré characteristic χ(φ(Fq [x]/(f(x)))) and count the num- ≤deg(p)/2 ber of high degree coe cients it agrees in with f(x). which is p(x) plus an error determined by the Frobe- Drinfeld module analogue of Lenstra’s algorithm nius trace tφ,p(x) of degree at most half that of p(x). The analogy with the Hasse bound is striking. The It is instructive to begin with Lenstra’s elliptic curve error in each case takes roughly half the number of integer factorisation algorithm before seeing its Drin- bits as the estimate. feld module incarnation. Pollard designed his p-1 algorithm to factor an integer that has a prime factor Factor degree by Euler–Poincaré characteristic modulo which the multiplicative group has smooth Gekeler’s bound concerns Drinfeld modules φ at an order. But this smoothness condition is rarely met. irreducible p(x). What happens at f(x) = i pi (x), Lenstra recast the role of the multiplicative group with the additive group associated with a random elliptic curve. If the integer has a prime factor mod- ulo which the randomly chosen elliptic curve has

30 FEATURES smooth order, the algorithm succeeds in extracting and apply the Drinfeld action φr(α) on α. It is likely that factor. For a random elliptic curve, this smooth- that φr(a) is a zero at some but not all factors pi (x) ness condition is met with a probability depending of f(x) and gcd(φr(a), f) gives a random factorisation. sub-exponentially on the size of the smallest prime A brief outline of why this is the case follows. As with factor. Consequently, it is among the most popular groups, the order of an element divides the cardinal- algorithms for integer factorisation, particularly as ity of an Fq [x] module. That is, Ord(α) divides the an initial step to extract small factors. Euler–Poincaré characteristic of φ(Fq [x]/(f(x))). In fact, with high probability, Ord(α) equals the Euler– Pollard’s p − 1 algorithm Poincaré characteristic Fix a positive integer B as the smoothness bound and denote by m, the product of all prime powers Ord(α) = pi (x) + tφ,pi (x) . at most B. Given an N to factor, choose a positive integer a < N at random. Assume a is prime to N i for otherwise gcd(a, N ) is a nontrivial factor of N . If N has a prime factor p with every prime power If the factors on the right have/don’t have a linear fac- factor of p − 1 at most B, tor independently and roughly uniformly at random, am −1 = (ap−1)m/(p−1)−1 0 mod p =⇒ p | am −1 then the algorithm yields a random factorisation. This and gcd(am − 1, N ) is likely a nontrivial factor of N . is indeed the case! The running time is exponential in the size of B. For typical N , B needs to be as big as the smallest The factors on the right lie in the short intervals factor of N and thus the running time is typically Ii centred at pi (x) with interval degree bounded exponential in the size of the smallest factor of N . by deg(pi )/2. The Frobenius trace distribution as- sures a certain semi-circular equidistribution of otφv,epir(xin) tiengethrse, short interval Ii . Remarkably, pi (x) + factorisation patterns in the unlike short intervals Ii are unconditionally proven to be Lenstra’s algorithm random enough. In summary, for random φ, the fac- Lenstra’s elliptic curve factorization algorithm factors torisation patterns of each pi (x) + tφ,pi (x) is like that every N in (heuristic) expected time sub-exponential of a random degree deg(pi ) polynomial. Further, they in the size of the smallest factor p of N . A key insight are independent. of Lenstra was to substitute the multiplicative group (Z/pZ)× in Pollard’s p − 1 algorithm with the group The computation of Ord(α) dominates the runtime E(Fp ) of Fp rational points of a random elliptic curve and can be performed e ciently through linear E over Fp . The running time depends on the smooth- algebra. This is in stark contrast to integers, where ness of the group order |E(Fp )| for a randomly chosen E. T≤he2√HpasasnedbLoeunnstdraguparoravnetdetehsatthhaits ||E(Fp )| − nding the order of an element in the multiplicative 1)| algorithm (p + group modulo a composite appears hard. runs in expected time sub-exponential in the size of p as- suming ianhtehueriisnttiecrovnalt[hpe+pr1o−ba2b√ilipt,ypth+a1t a r2a√npd]omis integer + Drinfeld modules with complex multiplication: smooth. Our last algorithm is distinguished in that it samples from Drinfeld modules with complex multiplication. Drinfeld module analogue A Drinfeld module φ has complex multiplication if its In the ensuing Drinfeld module version, random ellip- endomorphism ring tic curves will be recast with random Drinfeld mod- ules to factor polynomials. As before, let f(x) ∈ Fq [x] EndFq (x)(φ) ⊗Fq [x] (Fq (x)) denote the polynomial to factor. Pick a Drinfeld mod- is isomorphic to a quadratic extension of Fq (x). A typ- ule φ at random and a random element α in the ical Drinfeld module has an endomorphism ring only Fq [x] module φ(Fq [x]/(f(x))). The order, Ord(α), of isomorphic to Fq (x). Complex multiplication is the α is the smallest degree polynomial b(x) ∈ Fq (x) that rare case where the Drinfeld module is more symmet- annihilates α, that is φb(α) = 0. Extract and divide ric than typical. As with reducing elliptic curves over away the linear factors of Ord(α), rational numbers at primes, Drinfeld modules can be reduced at irreducible polynomials. The reduction r(x) := Ord(α)/ gcd(Ord(α), xq − x)

FEATURES 31 is deemed supersingular if the endomorphism ring as the Hasse invariant lift. is noncommutative, and ordinary otherwise. Every To show the algorithm indeed works, it remains to Drinfeld module with complex multiplication has the demonstrate that with constant probability, our ran- remarkable feature that the density of irreducible dom choice of φ with complex multiplication yielding polynomials where it is supersingular is roughly half. a random factorization of f(x). By complex multipli- cation theory and Carlitz reciprocity, this probability To factor a given polynomial f(x), the strategy is is identi ed with splitting probabilities in a certain hy- to choose a random Drinfeld module with complex perelliptic extension of Fq (x), and duly bounded. The multiplication. Using explicit formulae, construct a overall runtime is dictated by the time taken to com- Drinfeld module φ with comple√x multiplication by pute the Hasse invariant lift. An intricate algorithm the quadratic extension Fq (x)( x − c ) with c ∈ Fq for this task is devised in [1] using a fast procedure chosen at random. Then attempt to separate out the to compute its de ning recurrence. Remarkably, the irreducible factors of f(x) where φ is supersingular runtime exponent matches the best known factori- from the ordinary. This likely results in a random sation algorithm and is comparable in practice to factorisation of f(x), which is recursively factored to the fastest existing implementations. In light of this, obtain the complete factorisation. thorough further investigation of Drinfeld module inspired polynomial factorisation is warranted! To separate the supersingular factors, we look to the Hasse invariant, an indicator of supersingularity. The Acknowledgements Hasse invariant hφ,p ∈ Fq [x]/(p(x)) of φ at an irre- This work was supported by the European Union’s ducible p(x) vanishes if and only if φ is supersingular H2020 Programme (grant agreement #ERC-669891). at p(x). For the chosen φ, we construct a polynomial that is a simultaneous lift of Hasse invariants at all FURTHER READING irreducible polynomials of degree at most that of f(x). The e cient construction of this lift relies critically [1] J. Doliskani, A. K. Narayanan, É. Schost, Drin- on a Drinfeld module analogue of Deligne’s congru- feld Modules with Complex Multiplication, Hasse ence due to Gekeler. The common irreducible factors Invariants and Factoring Polynomials over Finite of this lift and f(x) are precisely the irreducible fac- Fields, J. Symbolic Comput., to appear. tors of f(x) where φ is supersingular. The GCD of f(x) [2] A. K. Narayanan, Polynomial factorization over and the lift separates out the supersingular factors from the ordinary, as desired. nite elds by computing Euler–Poincaré charac- teristics of Drinfeld modules, Finite Fields Appl. 54 Hasse Invariants and Deligne’s congruence: (2018) 335–365. [3] A. Panchishkin and I. Potemine, An algorithm For a Drinfeld module φ with de ning coe cients for the factorization of polynomials using elliptic (gφ, ∆φ), we now construct the aforementioned lift modules, Constructive methods and algorithms in of Hasse invariants. Consider the sequence rφ,k (x) ∈ number theory, p. 117. Mathematical Institute of Fq [x] of polynomials indexed by k starting with AN BSSR, Minsk, 1989 (Russian). rφ,0(x) := 1, rφ,1(x) := gφ(x) and for m > 1, rφ,m (x) := gφ(x) q m−1 rφ,m−1(x) Anand Kumar Narayanan −(xq m−1 −x ) ∆φ(x ) q m−2 rφ,m−2(x ). Gekeler showed that rφ,m(x) is the value of the nor- Anand is a research sci- malized Eisenstein series of weight q m − 1 on φ and entist at the Laboratoire established Deligne’s congruence for Drinfeld mod- d’Informatique de Paris ules, which ascertains for any p of degree k ≥ 1 with 6, Sorbonne Université. ∆φ(x) 0 mod p that hφ,p = rφ,k (x) mod p(x). His research interests are in the algorithmic Hence rφ,k (x) is a lift to Fq [x] of the Hasse invariants aspects of number theory guided by applications in of φ at not just one but all irreducible polynomials cryptography, coding theory and complexity theory. of degree k . Further, rφ,k (x), rφ,k+1(x) are both zero Anand was born in India and in his spare time enjoys precisely modulo the supersingular p(x) of degree at wandering jungles and mountains. most k . Since a factor of f(x) is of degree at most n, take gcd(rφ,n (x), rφ,n+1(x))

32 FEATURES 1 CCoommppuutteerrss aanndd mMaatthheemmaattiiccss KEVIN BUZZARD Kevin Buzzard Mathematicians currently use computers to do tedious calculations which would be unfeasible to do by hand. AInbtshtreacfut.tuMraet,hceomualdticthiaenys bceurhreenlptilnyguuses ctoomprpouvteertshteoordeomtse,doiorutsoctaelaccuhlatsitoundsewnthsichhowwotuoldwbreiteunpfreoaosfisb?le to do by hand. In the future, could they be helping us to prove theorems, or to teach students how to write proofs? Mathematics from the future analysis, topology and so on. Were software like this Mathematics from the future ttoobbeeaaddoopptteeddbbyyaabbrrooaaddeerrcclalassssooffmmaatthheemmaatticiciaiannss, , wwee mmiigghhtt sseeee aa ffuuttuurree wwhheerreetthheesseessyysstteemmssssttaarrtt Take a look at the following piece of computer code. ttoo bbeeccoommeeuusseeffuullffoorraabbrrooaaddeerrcclalassssooffrreesseeaarrcchheerrss Take a look at the following piece of computer code. ttoooo.. lemma continuous_iff_is_closed IInn tthhiissaarrttiicclelewweewwililllsseeeeaannoovveerrvvieiewwooffwwhhyytthheessee ssyysstteemmss eexxisistt aanndd wwhhaatt tthheeyyaarreeccuurrrreennttlylyccaappaabblele {f : α → β} : ooff.. 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exact hf _ wTvwTktstsiMhIopiikattMshIopstnsthhhhpatpnuhnnruhnrhhoosaveeoeeoiiljejeomilomilsisueuteyuyuaslsdggaalwawtrsetrslrtrtrarawwkikheliheiioiaotatddalaspsffnneheerrkkhmmLtLrtii-a-daitipaotoanwonehoehtteorerhhtttwmrtmrenreenoeawoawpeepoeeteteMMaa,,nfnhuhnirnoirmoomot,dndnopoiaepiloaetfucbcbdwwaagagoi,onironrrnlvrnvibnimesbdmeoooasfdosfdhlilipidlsaoiluaonusaesmaetmiathibbpendundolihudolih..vdsdasaeeaceagdcsafgdasfaeTTvftokaftaguogulvtlivtoihehoephenpenmkeamnoRaoRrhLauerraecrcgneuoeogeuoeoeoupmoopoLoeefospcffspocfaomuftumeufttumehearhhfnrthfrdtttandto.naeo.ahnaeoehp.riepifenscfeorvscoonrveourerAorcrvtc.veftmueefretbirtheihlmecorlocobhrAaoochaol.latyo.letynooknloneokfhnaefhetTdTedpecnmghptecngtcecithiehnhsoecuehnslseuehhsloep;ofe,he;idfmae,midaeeacbsracbustremagtaegocceelhoctpelthabcnteebneetkhentdahentdukyeidyidiseritttscetniittccifiinhssofihesLascniowagLainiattegaattreerilayurelaylauieattlne“ietttotoe“iotostonosthmw:hdm:hetn=uidetun=uiwsuhabshamiwbamLs”taLtse”iathLdehriacsedrhcesedsseieotodlsstiapfolooapnfhoiahooanoLiaocnrcnsartfimcinsnaetofmacsnsotaen,datein,doaoeoalnooeeollddneeaaiylddnnrtefafysrt--ff.-. ““WWauiotqaacthcaicthauccocaqiaticcccmmoohhhoohnosoppsppsfasunfaunsssseeeeeeoemeaopmsppsptidt,iad,acscsttdmdmhhclclllslesieyaeiaeaoaoiiiipnbpnbeekekepctpctnenesasasttffuspuespmaemaceaceaaeeitaeetrfrfpsptlttlatewoeotatewogeorgrrwrwoor,errierieferftornrtdonomirdomieeonaonarnrtrntirtirrcnwocoanworamogbdrmogdbmyptmyptpeutpeutsmmpsmmfooafhooahhhlaelsaerslrliotiyottyoiateiaetcioetciose”nnsh”nnhhshsloortorrtoaroiaoiootgeotgehe)rehl)rcylcyafyatsf,ytsg,efefftoftoiaoeiaoeorroresrytressxtrsxarveaersdsvedueshdswdeuhweaiaiytytcemceetasemaeeotaseocreccrncrnraprrreaxpsenrexseioeihooeihpoinipfrninfkrarankmdarsamrnsosreinoseilogilogniurimepnurimepncstncnrstnrctoesctogemplegamplmeayymgyhytpgothtlpftdlfcpdtrycphtpryhshpahbsalebialeiiteaieiutieoeioeeu.ieaonees.eannehspntsehptsccronnlrOhondlrOhdtmiaarUtemitaaaUettsaftrs.hfrehhterhhnotdrsoehenoKdosoeeKoaeWamaeeammeyauemmfdceyauemufdceldtuldtowlstowonelsstaon.ptsaa.ptlsoarctlsorctar,rmdhttar,emdhtetetMyeeeteeMoyboieobnoiihoanavcihooamavcnomapmnsaopmgrstoesnegrstlpesnelpcceaeiucuceareiu,soupmonrd,spplmondhaptelrahaettearaewtfaunhwuuvnnehgourvnetgagotreag(traeeniah(wtirraehniaivhetitrnehvivevutnneevslvurtneiislreto.iaiececo.ihsiredccaihsirddmmoartIrdmcmuohssrteIcuohnssere,heonuere,nhauegenaawageecawaatamensscfantamhnssaflntthhadaattwnihludaawcnhhlkusschhtaosiscntcaoicntescocheeeotescIomheeotsIhhmsthshhvsmmihnnsvmmitcnunstttheehtcueottheehoeohhoaohohaoaacoscaaahoraarcscaaarolrriiwwrhttelekiwkwyssssshttetfekkyssssI--tfI--

FEATURES 33 or C++). A mathematical proof is also a sequence expressed concern that this part of the argument of statements following a precise syntax, which per- was not human-checkable. They have now all been forms a certain task when the statements are inter- formally veri ed by computer. preted within the logic and language of pure math- The system Coq was the system used in George ematics. This observation is more than an analogy Gonthier’s formal proof of the four colour theorem, and goes back to Kolmogorov. and Gonthier also led the team which in 2013 formal- Initially it was observed that certain kinds of mathe- ized (again in Coq) the proof of the Feit–Thompson matical proofs (namely, proofs using something called theorem that every group of odd order is solvable. “constructive mathematics”) really are exactly the These two theorems are of di erent natures. The same thing as certain kinds of computer programs. four colour theorem is another example of a theo- As a consequence, the tools developed by computer rem where the standard argument ends up having scientists to verify that code has no bugs can also to go through thousands of tedious case checks, but be used to show that a proof in constructive mathe- this is not what the Feit–Thompson proof looks like; matics has no gaps or errors. the latter proof is a long group-theoretic argument However, often mathematicians do not work purely for which Thompson recieved a Fields Medal in 1970. in constructive mathematics. One key di erence The modern mathematical proof of the odd order between a proof and a program is that proofs are theorem is written up in two books by Peterfalvi and allowed to use non-constructive things such as the Bender–Glauberman, both published in the LMS lec- axiom of choice, which can be viewed in this optic ture note series [1, 3]. The Coq proof was created by as a function for which there is no algorithm, i.e., a team of 15 people over a period of 6 years, during no program. But some of the newer computer proof which time they formalized the 328 pages of research checking tools (such as Lean) have been designed level mathematics in the two books as well as all the with mathematics in mind, and have simply added prerequisites. Perhaps surprisingly, over two thirds things like the axiom of choice as an axiom of the of the code base is the prerequisites for the books system. This means that tools like Lean can now rather than the books themselves. The prerequisites check “normal” mathematical proofs. In short, if you covered basic results in the representation theory can prove something using classical logic and ZFC set of nite groups, Galois theory, some commutative theory (which is what most pure mathematicians just algebra, and a little number theory. Moreover, most call “mathematics”) then you can, in theory, prove it of the people involved did not work anywhere near in Lean. the full six years that the project lasted. Ultimately there were under ve lines of computer code corre- What mathematics is formalized? sponding to each line of the book on average, a gure What however can we prove in practice in Lean or in which perhaps better re ects the e ort, although it these other systems? Which results are already avail- should be said that writing a correct line of code in able to us? Let us start at the top, and go through one of these languages can take a long time. some of the traditional breakthrough results. The work of Hales et al., and Gonthier et al., are hence The systems HOL Light and Isabelle/HOL were used evidence that these computer systems are now capa- in the formally veri ed proof of the Kepler conjec- ble of handling very long and complex proofs about ture (“HOL” stands for Higher Order Logic). The for- relatively simple objects such as nite groups or mal veri cation was done by a team led by Tom spheres. Hales. Hales’ 2017 talk at the Big Proof workshop In 2016, Ellenberg and Gijswijt established a new at the Newton Institute is available on the INI web- upper bound for the cap set problem, resolving the site (see tinyurl.com/y4avuzj9) and tells the interest- cap set conjecture. The cap set conjecture is a com- ing story of how he ended up having to formalize binatorial question about the growth of the size of the paper proof which he and Ferguson had con- a certain sequence related to an n-dimensional ver- structed in the late 1990s. The argument involved, at sion of the card game “Set”, as n increases. The some point, checking thousands and thousands of Ellenberg–Gijswijt proof was published in the Annals nonlinear inequalities, and the human referees had of Mathematics. In 2019 Sander Dahmen, Johannes Hölzl and Rob Lewis formalized the proof in Lean. Dahmen is a number theorist; Hölzl’s background is in computer science and Lewis’s is in logic. The mathematical proof was remarkable in the sense

34 FEATURES that many of the methods used were elementary, degree. Formalising undergraduate level mathematics making the result a very good candidate for formali- is a very di erent ball game. It takes far less time, sation. The Lean proof is modern mathematics being and can be worked on by undergraduates who are formalized in a realistic time frame. learning about the subject for the rst time. All of The next example I will speak about here is of a the systems I have mentioned so far contain a lot di erent nature; it is the work of Sébastien Gouëzel of undergraduate-level mathematics, but one cannot and Vladimir Shchur from [2]. The Morse Lemma easily port theorems from one system to another is a result about the Hausdor distance between a (the problem is analogous to porting computer pro- geodesic and a quasi-geodesic with the same end- grams from one language to another), so when new points in a Gromov-hyperbolic space. In 2013 Shchur systems come along these libraries need to be built gave a quantitative (i.e., with explict constants) proof from scratch. Each system contains some, but not all, of the lemma, which was published in the Journal of the pure mathematics component of a typical UK of Functional Analysis. Gouëzel decided to formalize undergraduate degree; the union of all the systems the proof in Isabelle/HOL, and during the process he might well contain most of it nowadays, however found a serious error missed by both author and because of the portability issue no one system has referee. A completely new argument was found by access to all of it. Gouëzel and Shchur to x the problem, and Gouëzel I started learning Lean by formalising the questions formalized the new proof in Isabelle/HOL. The result and solutions in Imperial’s rst year “introduction to was written up as a joint paper which was published proof” course, a course which I was lecturing at the in the Journal of Functional Analysis earlier this year. time. I typically got stuck several times per question, Again this is modern mathematics being formalized I asked hundreds of questions in the Lean chat- in a realistic time frame, but this time the objects room on Zulip and they were answered promptly and are more complex. politely, often by computer science PhD students and Last year I got interested in exactly how complex the post-docs. When I had got the hang of things a little objects handled by this sort of software could be. more, I started encouraging Imperial’s undergradu- Perfectoid spaces were discovered by Peter Scholze ates to join in, and we have made good progress. in 2011 and his applications of the theory to fun- Over the last two years, mathematics undergradu- damental questions in the Langlands program and ates at Imperial have been formalising results which arithmetic geometry won him a Fields Medal in 2018. they have been taught during their degree, or learnt Earlier this year Patrick Massot, Johan Commelin and independently. For example the theorem of quadratic myself formalized the de nition of a perfectoid space reciprocity, Sylow’s theorems, the fundamental theo- in Lean. The formalisation process took around ten rem of algebra, matrices, bilinear maps, the theory months in total, and we wrote well over 10,000 lines of localisation of rings, the sine, cosine and exponen- of code in order to formalize this de nition, involving tial functions, tensor products of modules and many (for example) formally proving many of the theo- other things have been implemented by Imperial’s rems in Bourbaki’s “Topologie Générale”. We have mathematics undergraduates in Lean. Chris Hughes as yet proved nothing at all about perfectoid spaces and Kenny Lau in particular are two students who and furthermore cannot yet produce any non-trivial have formalized many results from our undergrad- examples, because we still need to formalize the gen- uate curriculum. There is more to come; we are eral theory of discrete valuations on nite extensions working on the Galois theory course and after that of the p-adic numbers. Formalising this theory would we will start on the representation theory course. be an ideal problem for a PhD student who wanted One possible future project would be to turn some to learn the material in an innovative way, however of these formalisation projects into some sort of it would be far less e cient in terms of time. formally veri ed book or website; imagine a book on Galois theory which was guaranteed to contain no Undergraduate level formalized mathematics errors, and which every proof which the reader did As well as these “complex” formalisations, there are not follow could be “unfolded” more and more until formalisions of much of the pure material which is the penny dropped. typically taught in a UK undergraduate mathematics In 2018–19 I also supervised BSc student Anca Ciobanu, who formalized the basics of group coho- mology in Lean, and MSc student Ramon Mir who formalized Grothendieck’s notion of a scheme in

FEATURES 35 Lean. I was very surprised to discover that schemes just as knowledge of LaTeX now is. Another purpose had not been done in any other formal system. As is an ongoing project to completely digitise the pure a result it seems that Mir’s work is of a publishable mathematics part of Imperial’s undergraduate cur- standard. Nowadays we see students typing up their riculum. lecture notes in LaTeX; it would be wonderful to see, I have also used Lean for outreach, showing in the future, students typing up their course notes schoolkids how to prove things like 2 + 2 = 4 from in Lean or one of these other systems. the axioms of Peano arithmetic and then raising the Using Lean for teaching. question of how much more mathematics can be done like this. In the 2018–19 academic year, I taught Imperial’s Using Lean to help with research? introduction to proof course again, this time using Lean live in the lectures for some of the proofs. For Currently, the only thing that Lean can o er the example, I proved that a composite of surjective func- researcher is a “certi cate” that any theorem they tions was surjective in Lean, I proved various results have formalized in Lean is indeed correct. Many about equivalence relations in Lean, and I proved humans feel that they do not need this certi cate. I Cantor’s diagonal argument in Lean. I wrote the code live during the lectures, projecting my laptop onto nd it interesting that more and more mathematics the screen, but I also circulated “traditional” proofs relies on unpublished work; talk to any pure mathe- in the course notes afterwards. All of this was done matician in any big area and they will know of some under the watchful eye of Athina Thoma, a specialist theorems which rely on results in the “grey literature” in mathematical education, who interviewed and sur- (a letter or an email from X to Y) or even results veyed the students and is writing up her conclusions which are not in the literature at all (“a forthcoming for a forthcoming publication with Paola Iannone, paper”, sometimes announced years ago and which another such specialist based in Loughborough. This never came forth). Humans are good at forgetting approach was completely new to both me and the to go back and ll in these details. Lean will not students, and I note with interest that their perfor- let you forget. The computer-generated red squiggly mance on the (conceptually complicated) question line underneath your theorem reminds you that your about binary relations on the end of year exam was proof is not complete, even if your work relies on done very well. Of course one has to be careful here; results announced by others and hence “it is not not all the students will want to see mathematics your job to complete it”. During our work formalizing being done with computers, and of course these perfectoid spaces, it got to the point where I knew students should not be disadvantaged by such an “we had done it”, but until the de nition compiled we approach. However I did think it was important that knew we had not convinced Lean. Another issue is the students learnt, as early as possible, what a logi- the fact that formalising the proof of a big theorem cal argument looked like, and how a logical argument like Fermat’s Last Theorem, given the state of the is something which can be explicitly checked by an art, would surely take more than 100 person-years. algorithm. This is not how mathematics is taught in Getting money for this would involve a grant proposal schools but it seems to me to be a crucial message which no grant agency would fund, and which would which an introduction to proof course in a mathe- yield a result which no mathematician would care matics department should convey. Patrick Massot is about anyway — we all know Fermat’s Last Theorem also using Lean in the mathematics department at is true, because we know the experts have accepted Orsay, in his introduction to proof course. the result. On Thursday evenings during term time I run a Lean Because Lean is o ering something that pure math- club at Imperial called the Xena Project (with asso- ematics researchers do not generally want (a fully ciated blog at xenaproject.wordpress.com), where formalized veri cation of their theorems) at a price undergraduates learn to formalize the mathematics which researchers generally will not pay (the time which they are learning in lectures, and any other taken to learn how to use the software, and then the mathematics of interest to them. One purpose of the time taken to formalize their own proofs), in practice project is to make knowledge of formalisation more one has to rule this out as a viable practical applica- prevalent amongst the undergraduate community, tion of these tools at this point in time. Things might

36 FEATURES change in the future, but rather than speculate on computer scientists what we are actually doing, in a how good computers can or will become at under- format which is more useful to them than tens of standing natural language (i.e., arXiv preprints), let thousands of arXiv preprints where we use words me simply observe that currently they are nowhere like “normal” and “complete” to mean ve di er- near good enough, and instead turn to other things. ent things and where we write in the idiomatic and Tom Hales has suggested a completely di erent use bizarre English language, thus adding another layer of these tools, and has won a multi-million dollar of obfuscation to the material. Formalisation solves Sloan grant to implement his idea, which is called this problem — it makes us say what we mean in a FABSTRACTS (Formal Abstracts). Hales suggests that clear and coherent manner. This is what these people we should create a mathematical database containing need to make tools for us — and we are not giving not proofs but formalized statements of theorems, it to them. Mathematicians do not seem to have any cross-referenced to the traditional literature (journal desire to formalise a proof of Fermat’s Last Theorem, articles etc). He proposes that Lean be used for this however computer scientists are desperate to train database, and indeed that is why I chose Lean. The their AI’s on a formalised proof, to see how much reason this idea is a game-changer is that if all the further they can take things. If any mathematician de nitions are there, then formalising a statement reading this decides that they would be interested in of a new theorem can be easy. The fact that Lean taking up the challenge of formalising the statement can handle the de nition of a perfectoid space is of one of their theorems in Lean, then they can nd surely evidence that it can handle any pure mathe- me at the Zulip lean chat. matical de nition which we can throw at it. There Computer scientists do not understand a lot of what will also be a natural language component to the we as mathematical researchers do, or how we do it. project; the current plan is that theorems will be Let’s tell them in their own language, and see what presented in a controlled natural language, i.e., in happens next. Computer scientists have developed human-readable form, understandable to a mathe- tools which can eat mathematics, but until these matician with no Lean training. One can imagine in tools get more self-su cient they will need to be the near future such a database being constructed, hand-fed. It will be very interesting to see what these initially concentrating on the areas of speciality of tools become as they grow. the mathematicians already involved. What would be the potential uses of such a database? FURTHER READING The rst obvious use would be search. Say a [1] H. Bender and G. Glauberman, Local Analysis researcher is wondering what is known about a math- for the Odd Order Theorem, Cambridge University ematical statement. They formalize the statement, Press, 1994. look it up in the database, and get either a perfect hit [2] S. Gouezel and V. Shchur, A corrected quanti- (a reference to the literature) or perhaps some partial tative version of the Morse lemma, J. Funct. Anal. results (several references, perhaps to variants of 277 (2019) 1248–1258. the statement proved under certain hypotheses). If [3] T. Peterfalvi, Character Theory for the Odd Or- we can train PhD students and post-docs to learn der Theorem, Cambridge University Press, 2000. enough about Lean to state their results, then this database could grow quickly. Kevin Buzzard The second obvious use would be AI. Already com- puter scientists are trying to develop tools which Kevin Buzzard is an can actually do mathematics better than humans. algebraic number theo- They want to train their AI on databases — but rist and a professor of these databases are few and far between, and sev- pure mathematics at Im- eral of the databases which are used in practice are perial College London. databases of solutions to gigantic logic puzzles which He tweets occasionally bear no relation to modern mathematics. I believe at @XenaProject. that it is up to us as a community to explain to

FEATURES 37 Reciprocal Societies: Belgian Mathematical Society The Belgian Mathemat- journal called the Bulletin de la Société Mathématique ical Society (BMS) was de Belgique. The Bulletin underwent several transfor- founded on 14 March 1921 mations in the following decades and merged with (π-day!) at the Free Uni- the journal Simon Stevin in 1994. It is now addressed versity of Brussels; such to a generalist audience, with a broad editorial board a choice of date is per- covering a wide range of elds. Since 2003, it is avail- haps an early testimony able electronically via Project Euclid and all issues to the celebrated Belgian of the new Bulletin older than ve years are made sense of surrealism. Of the nine founders of the available with free access. society, the best known are Théophile De Donder, Attendance at the monthly meetings eventually dwin- Lucien Godeaux and Alfred Errera. The aim of the dled and the habit was abandoned in the seventies. society was the same in those days as it is now: In the eighties an annual meeting was launched, with great success at rst. However, after a few years, to contribute to the development and di u- attendance diminished as the pertinence of such a sion of all forms of mathematics in Belgium. non-specialized national meeting found less priority [The BMS] is concerned with mathematics, in the already hectic lives of academics. Since those pure and applied, in the broadest sense. It days, the BMS has diversi ed the array of mathemat- will try to establish a permanent link between ical events in which it takes part: co-organizing joint secondary school and university. meetings with other National Mathematical Societies (including a joint meeting with the London Mathe- At the beginning, and matical Society organized in Brussels in 1999); orga- over the course of sev- nizing joint meetings with the high school teachers’ eral decades, the society associations; organizing the bi-annual “Ph.D. Day” organized monthly meet- during which Ph.D. students or young researchers ings during which an have the opportunity to present their work through extremely varied array posters and short communications; “thematic after- of subjects were con- noons” during which the works of primed mathemati- sidered. Indeed, we nd cians (Fields medalists, Abel prize winners, . . . ) are lectures on mathemati- explained broadly by experts of the di erent elds. cal physics (with e.g. De The BMS also funds the Lucien Godeaux prize and is Donder), astrophysics a sponsor of several mathematical initiatives aimed (Lemaître), algebraic ge- A founding father Lucien at younger audiences. ometry (Godeaux), analy- Godeaux (engraving by J. On π-day 2021 our Society will celebrate its 100th sis (De la Vallée Poussin, Bonvoisin 1947, courtesy anniversary. The various mutations that it has gone Lepage), engineering of the Royal Library of through were necessary and helped the Society to (van den Dungen), math- Belgium). serve the mathematical community over time and remain relevant to the 21st century Belgian mathe- ematics of insurance, and secondary school math- matical community. ematics (with A. Mineur, sharply remembered by generations of Belgian school children for his treatise Yvik Swan on descriptive geometry). Amongst foreign speakers, BMS President let us just pick two curios. On top of various lectures on integration, Lebesgue gave a talk in 1925 on ruler Editor’s note: the LMS and the BMS have a reciprocity and compass constructions. In 1922, Millikan gave agreement meaning members of either society may a lecture, comparing his ideas on the electron with bene t from discounted membership of the other. those of Planck and De Donder. From 1947 onwards, and under the impetus of topol- ogist Guy Hirsch, the BMS started editing its own

38 EARLY CAREER RESEARCHER Microtheses and Nanotheses provide space in the Newsletter for current and recent research students to communicate their research ndings with the community. We welcome submissions of micro- and nano-theses from current and recent research students. See newsletter.lms.ac.uk for preparation and submission guidance. Microthesis: Hypergraph Saturation Irregularities NATALIE C. BEHAGUE A graph is F -saturated if it doesn’t contain a copy of a graph F but adding any edge creates a copy of F . The maximum number of edges an F -saturated graph can have is well-studied and called a Turán number. Our topic here is the minimum number of edges an F -saturated graph can have, which behaves quite di erently. What is saturation? Asymptotics A graph is F -free if it does not contain a copy of Turán’s Theorem and the Erdo´´s–Stone Theorem tell the graph F . Given a xed graph F and a positive us that for a (non-empty) graph F , integer n, Turán’s extremal number, ex(F, n), is the maximum number of edges an F -free graph G on n ex(F, n) = 1− 1 − 1 + o (1) n , vertices can have (see gure 1). What if we were to χ(F ) 2 ask about the minimum number of edges? If we just replace the word ‘maximum’ with ‘mini- where χ(F ) is the chromatic number of F . In partic- mum’ in the de nition above we get a trivial function, ular, ex(F, n converges to a limit as n tends to since a graph with no edges will be F -free. But notice in nity. n)/ 2 that in any maximal F -free graph, adding any edge creates a copy of F . This inspires the following de - Can we say anything similar about the satura- nition: we say a graph G is F -saturated if it does not tion number? Kászonyi and Tuza [2] proved that contain a copy of F as a subgraph but adding any sat(F, n) = O (n). Tuza [5] went on to conjecture that edge creates a copy of F . sat(F,n) Now we have an equivalent de nition of Turán’s for every graph F the limit limn→∞ n exists. extremal number: ex(F, n) = max{e (G ) : |G | = n and G is F -saturated} Forbidden families and if we replace max with min we get the saturation The de nition of saturation can be extended to fam- number: ilies of graphs. For a family F of graphs (called a sat(F, n) = min{e (G ) : |G | = n and G is F -saturated}. forbidden family), a graph G is F-saturated if it does not contain any graph in Fas a subgraph, but adding ex(F, n) = n2 sat(F, n) = n − 1 any edge creates a copy of some graph F ∈ F as a 4 subgraph of G . We de ne the saturation number in the same way as before: F= sat (F, n) = min{e (G ) : |G | = n and G is F-saturated}. For a family of graphs F we have sat (F, n) = O (n) Figure 1. Extremal and saturation numbers for the triangle [2], just as we did for single graphs. However, the generalisation of Tuza’s conjecture to nite families For example, if F is the triangle then the maximal of graphs is not true, as shown by Pikhurko in [4]: tthrieanmglien-imsaatul rtaritaendglger-aspahtuhraatsedn4g2rapehdgiessa. has only n − 1 edges. In contrast, star, which

EARLY CAREER RESEARCHER 39 Theorem 1. There exists a nite family F of graphs contains two types of r -graph: a pair of intersect- such that sat (F, n)/n does not tend to a limit as n ing complete r -graphs; or some disjoint complete tends to in nity. r -graphs together with a ‘bridge’ edge between them. In Pikhurko’s construction, the graphs in F depend Figure 2. The family F of r -graphs for r = 5 and k = 7 on some xed constant k . For n divisible by k , one can construct an F-saturated graph on n vertices An example can be seen in gure 2, where the ver- that uses relatively few edges. For n not divisible tices surrounded by a blue line represent a copy of by k , there is no such ‘nice’ construction and an F-saturated graph on n vertices is forced to con- tain many extra edges. Pikhurko asked whether a similar construction was possible for families of hypergraphs. Saturation for r -graphs the complete r -graph on k vertices aKkb(rr)i,dagnedevdegreti.ces grouped by a black line represent An r -graph H , often referred to as an r -uniform Tuza’s conjecture itself remains open. It seems di - hypergraph, is a pair, (V, E), of vertices and edges cult to reduce the family above to a single hypergraph where the edge set E is a collection of r -element while keeping the non-convergent behaviour: I have subsets of the vertex set V . Note that a 2-graph is only been able to reduce the family to one of size exactly a graph. four. It could be that such a large gap in the asymp- totics is not possible for a single hypergraph, and The de nitions of saturation and saturation num- one must look at the coe cients. bers transfer immediately to the context of r -graphs — the only di erence is that the edges contain r FURTHER READING vertices rather than 2. [1] N. C. Behague, Hypergraph Saturation Irregu- larities, Electron. J. Combin. 25 (2018) 2–11. For a family F of r -graphs, it was shown by Pikhurko [2] L. Kászonyi, Z Tuza, Saturated graphs with min- [3] that sat (F, n) = O nr −1 when the family con- imal number of edges, J. Graph Theory 10 (1986) tains only a nite number of graphs. This leads to 203–210. the following generalisation of Tuza’s conjecture to [3] O. Pikhurko, The minimum size of saturated r -graphs, rst posed by Pikhurko [3]. hypergraphs, Combin. Probab. Comput. 8 (1999) 483–492. Conjecture 1. For every r-graph F the limit [4] O. Pikhurko, Results and open problems on sat(F,n) minimum saturated hypergraphs, Ars Combin. 72 limn→∞ nr −1 exists. (1994) 111–127. [5] Z. Tuza, Extremal problems on saturated As in the 2-graph case we can further generalise this graphs and hypergraphs, Ars Combin. 25 (1988) conjecture by replacing the single r -graph F with a 105–113. nite family of r -graphs F. Pikhurko’s question now Natalie Behague becomes “Does there exist a family F of r -graphs such that sat (F, n)/nr −1 does not converge to a limit Natalie Behague is a as n tends to in nity?” I have shown in [1] that the PhD student at Queen answer to this question is yes. Mary University of Lon- don under the supervi- Theorem 2. For all r ≥ 2 there exists a family F of sion of Robert Johnson, r -graphs and a constant k ∈ N such that researching problems in extremal combinatorics. She enjoys boardgames, sat (F, n) = O (n) if k | n baking, and playing tag rugby quite badly. if k n  nr −1 Ω  In particular, sat (F,n) does not converge. nr −1 The proof uses a similar approach to Pikhurko’s construction for graphs. The forbidden family used

40 REVIEWS The Princeton Companion to Applied Mathematics by Nichloas J. Higham (editor), Princeton University Press, 2015, £77, US$ 99.50, ISBN: 978-0691150390 Review by David I Graham The rst question that to be generally good. In terms of self-containedness, comes to mind when some of the contributions in the later parts require reviewing a book like prior knowledge not fully detailed in the introductory this is “Why?” A Google parts, though more detailed investigations utilising search for “Applied Math- the reference lists should mean an interested reader ematics” nds 382 mil- would be able to ll the gaps. lion pages related to the Returning to the material covered, Part IV is really subject, many of which the heart of the book. It describes in considerable will be very detailed and detail (over 400 pages), forty “Areas of Applied Math- with access to anima- ematics” including straightforward choices such as tions and relevant computer code as well as links to various avours of mechanics, di erential equations related work. The editors are, of course, well aware and numerical methods but also less obvious areas of this and try to answer the question themselves. In such as algebraic geometry. There is a very readable the Preface they claim that the distinguishing feature contribution on “Symmetry in Applied Mathematics\", of the book is that it is “self-contained, structured which starts from the symmetries of plane gures reference work giving a consistent treatment of the and goes as far as symmetry breaking, with much subject”. discussion related to the various symmetries seen in The book is in eight parts and runs to almost 1000 Taylor–Couette uid ow between co-rotating cylin- pages. There is certainly a serious attempt to be self- ders. The author confesses that the article “barely contained, with the rst part containing the longest scratches the surface”. A typical example for this articles in the book, de ning basic language and part is the ten page contribution on “Fluid Dynamics”, terms from coordinate systems through calculus up which rattles along at great pace, covering everything to operators and stability. The second part then from 2-dimensional streamlines through ight aero- brie y reminds the reader of essential concepts from dynamics up to ow instability — enough material to asymptotics to wave phenomena. These are arranged in alphabetical order, occasionally meaning that there ll a decent course module. Similarly, the nineteen is no natural ow from one contribution to the next. page contribution on “Numerical Linear Algebra and The structure becomes more free-form as we get Matrix Analysis” is a comprehensive collection of the further into the book. This is inevitable given the main results relating to matrix computations and great variety in topics covered. With over 160 authors notes some useful ‘must have’ references. I have to contributing to the articles, consistency was always confess that some of the other contributions are going to be di cult to achieve and there is consider- rather dry for my taste, though the articles generally able variation in the later parts, especially in the level represent excellent starting points for further inves- of detail and follow-up information. One contribution tigation — which is one of the great strengths of the in Part V has a reference list of one item, namely a book as a resource. book written by the contributor himself. Conversely, Parts V (“Modelling”), VI (“Example Problems”) and the exemplary contribution on nancial mathematics VII (“Application Areas”) together give us 64 di erent in Part V provides not only an excellent reference examples, averaging at around ve pages in length. list but also a discussion of what to look for in each Several of these — including a contribution on “Sport” of the references. Overall, the reference lists seem that strangely covers only sailing, rowing and swim-

REVIEWS 41 ming — are related to uid dynamics, meaning that My nal thoughts are that this book is an excellent there is some repetition between articles. The article resource for any mathematics departmental library. on “A Symmetric Framework with Many Applications” The articles cover a vast array of di erent applica- outlines a nice example of a unifying viewpoint for tion of mathematics and are generally well-written minimisation problems of various kinds. As a com- with useful reference lists. In particular, the book rep- putational applied mathematician, I welcomed the resents an excellent launch point for individuals such contributions in Part VII related to historical develop- as project students looking for an area of Applied ments in programming languages (and the confusion Mathematics to investigate in greater depth. to which di erent conventions can lead) and the future in terms of high-performance computing. David Graham A particularly interesting feature of the book is the last Part (VIII), which o ers some “Final Perspectives”. David is a Senior Lec- I found many of these contributions to be extremely turer in Applied Mathe- thoughtful and useful, including advice on how to matics at the University read Mathematics articles or to write articles or even of Plymouth. His main general interest books. As an author of reasonably research interests are large fortran codes myself, I found the sections on in developing and using “Reproducible Research” and “Experimental Mathe- numerical methods for matics” to be especially thought-provoking. uid dynamics. David is a keen footballer and, in decreasing order of competence he also plays guitar, banjo, ukulele and bouzouki. The Turing Guide by Jack Copeland, Jonathan Bowen, Mark Sprevak and Robin Wilson, Oxford University Press, 2017, £19.99, US$ 29.95, ISBN: 978-0198747833 Review by David Glass For several decades and work, The Turing Guide is an excellent contribu- after his tragic and tion to this development and the growing literature untimely death in 1954 on Turing. at the age of just 41, very The book consists of forty-two chapters divided little was known about into eight sections, with the rst section providing Alan Turing’s important biographical material. The rst chapter provides a work at Bletchley Park brief sketch of Turing’s life, with a helpful timeline during the second world of key events, while in the second chapter, entitled war. However, as the ‘The man with the terrible trousers’, Turing’s nephew, full scale of his achieve- Sir John Dermot Turing, provides a unique family ments in codebreaking perspective. The third chapter is a compilation of and the relevance of the wartime e ort to the history extracts from papers and reminiscences of the late of computing became clearer, Turing’s reputation Peter Hilton, who worked with Turing for the last 12 has increased dramatically. By providing a very wide- years of Turing’s life. According to Hilton, Turing was ranging and yet accessible account of Turing’s life

42 REVIEWS a ‘warm, friendly human being’, ‘an approachable paranormal (Chapter 32), which provides context to and friendly genius’ and he gives a sense of Turing’s Turing’s equally surprising comments about the pow- ‘quirky and infectious sense of humour’. In addition ers of extra-sensory perception in his famous article to providing insights into Turing’s early life, school- in Mind in 1950. days, and working life, the rst section also discusses Sections six and seven look at Turing’s contribu- his conviction for homosexuality, the terrible chem- tions to biology and mathematics, with the former ical treatment he had to undergo as a result, the providing an introduction to his fascinating work on subsequent apology from Gordon Brown in 2009 and morphogenesis in which he used reaction-di usion royal pardon in 2013, as well as the circumstances of equations to model the development of biological his death. As for whether Turing committed suicide, structures. In Chapter 33, Margaret Boden explores ‘We shall most probably never know.’ the in uence of Turing’s work on research in self- Those interested in the history of computing and organisation, arti cial life, and structuralism in biol- the codebreaking work at Bletchley Park will nd ogy. Interestingly, she argues that Turing’s innovative a lot of useful material in sections two to four. In ideas were ‘out of line with the biological orthodoxy’ Chapter 6, Jack Copeland provides an introduction of neo-Darwinism and associates his approach with to Turing’s greatest contribution: the universal Turing that of the structuralist D’Arcy Thompson. In Chap- machine. He also addresses the origin of the concept ter 35, Bernard Richards describes how he carried of the stored program computer, an issue which is out work to validate Turing’s theory by showing how further explored in Chapter 20 in the context of the solutions to his equations in three dimensions could development of Baby, the rst electronic universal explain the shapes of marine creatures known as stored program computer, in 1948. Chapters 7 and 37 Radiolaria. He had arranged to meet Turing on 8 June provide very readable introductions to Turing’s impor- 1954 to show him his intriguing ndings, but sadly tant work on Hilbert’s famous Entscheidungsproblem, Turing died the previous day. while Chapter 24 explores congruences between Tur- Chapter 36 provides a very helpful overview of Tur- ing’s work and the earlier ideas of Charles Babbage ing’s work on the central limit theorem, group theory and Ada Lovelace. and the Riemann hypothesis, while other chapters in The Enigma machine is the subject of Chapter 10 with this section explore further topics such as Turing’s Turing’s bombes (special purpose electromechanical work on randomness, which ‘anticipated by nearly computers used to decipher Enigma messages) the thirty years the basic ideas behind the theory of subject of Chapter 12. Both of these chapters pro- algorithmic randomness’. In Chapter 38 (and also 13), vide helpful explanations about how the machines Edward Simpson, who worked as a cryptanalyst at worked while Chapter 14 explores Tunny, the encryp- Bletchley Park (and after whom Simpson’s paradox tion machine used by Hitler later in the war. Turing’s is named), provides a very interesting and insightful contribution, as well as that of Bill Tutte, Max New- explanation of Banburismus, a process for reducing man and Tommy Flowers, including the essential role the work the bombes had to do to decipher Enigma of Flowers’ Colossus computer, are all discussed. A messages. He also gives a nice account of Turing’s number of people who worked as codebreakers or use of Bayes factors and logarithmic scoring to weigh operated Colossus or the bombes provide fascinating evidence. accounts of their time at Bletchley Park. The nal section has a very interesting chapter Section ve explores Turing’s contributions to arti - on whether the universe is computable. This wide- cial intelligence and the mind. These chapters include ranging chapter covers topics such as computability, discussions of topics such as the Turing Test, his Turing’s thesis (or the Church–Turing thesis) and mis- use of heuristic search techniques in the bombe, his conceptions about it, and the physical computability early work on connectionist models, child machines, thesis. It ends by arguing that Turing was not com- his contribution to cognitive science, and his chess mitted to the idea that the universe or even the brain program. Particularly interesting is Diane Proudfoot’s is computable. The nal chapter highlights Turing’s discussion of Turing’s concept of intelligence (Chap- impressive legacy both within science and beyond. ter 28). She argues that Turing thought of intelligence The Turing Guide is extremely informative, highly read- as an emotional concept that is partly in the eye of able, and well produced with many photographs and the beholder and she also argues against the com- useful gures to aid exposition (though some colour mon view that Turing was a behaviourist. Perhaps the most surprising contribution is a chapter on the

REVIEWS 43 would have been nice). The preface states the book David Glass was ‘written for general readers, and Turing’s sci- enti c and mathematical concepts are explained in David Glass is a senior an accessible way’. This has been achieved with lecturer in the School great success. However, those working in a range of Computing at Ulster of elds will also bene t a lot from articles written University. He carries by experts and pointers to the extended literature. out research in arti cial In other words, it does exactly what a good guide intelligence, philosophy should do. Alan Turing’s legacy has grown enormously of science, and the mod- in recent years and The Turing Guide will serve to elling of complex systems. He enjoys football and further enhance understanding of his remarkable rugby (watching not playing) and has an interest in achievements. the relationship between science and religion. Anxiety and the Equation: Understanding Boltzmann’s Entropy by Eric Johnson, MIT Press, 2018, £18, US$ 22.95, ISBN: 978-0-26203-8614 Review by Andrew Whitaker Nobody today would to an enormous number or microstates, which are question that Ludwig particular states of the individual particles. Boltzmann was one of The second set of ideas is related to the famous the greatest of physi- Boltzmann distribution. Given a xed amount of cists. It could be said energy to be divided between a number of particles, that he provided the this distribution tells us, with good probability, the route from nineteenth- number or particles having various values of energy century physics, which at equilibrium. was broadly macroscopic in nature, and where The last idea clari ed the idea of entropy. While in the atoms were treated with nineteenth century this was looked on as a macro- considerable suspicion scopic function of the thermodynamic variables of (at least by many physicists, less so by chemists, the system, for example the entropy of a gas could but even more so by a number of philosophers) and be expressed as a function of its pressure and vol- on to a physics largely based on atoms and their ume, for Boltzmann its de nition was statistical. The constituents, with theoretical ideas centred on the relevant and famous equation is S = k logW , where quantum. His main work used three considerably S is the entropy of the system, W is the multiplicity overlapping sets of ideas. The rst was the existence of the appropriate macrostate, and k , of course, is of atoms and their centrality in any discussion of always known as Boltzmann’s constant. The idea physics. In particular this implied the extensive use of this equation is undoubtedly due to Boltzmann, of probability arguments, and especially the idea that who discussed all the ideas at great length, though the macrostate, or macroscopic state of the system Eric Johnson points out that the actual equation and that we actually meet, will be one corresponding

44 REVIEWS indeed ‘Boltzmann’s constant’ itself were in fact writ- over the two halves of a lecture room, atoms over ten down by Max Planck! the two halves of a volume. He divides 7 units of The enormous importance of Boltzmann’s work, so energy over 4 atoms and calculates the resulting dis- obvious today, was unfortunately not at all clear tribution. He studies entropy by analysing di erent to a number of extremely in uential scientists and numbers of bedbugs moving between the two halves philosophers, and disagreements were expressed of an auditorium. For just 4 bedbugs the entropy increasingly aggressively from around 1890, when initially moves towards what might be called an ‘equi- Boltzmann was in his mid-forties. The three chief librium value’ but clearly that terminology is hardly antagonists were Ernst Mach, Planck and Ernst Zer- appropriate because it continues to uctuate wildly melo, who was Planck’s assistant. about this value. For 1000 bedbugs we see similar As an arch-positivist, Mach was diametrically behaviour but the uctuations are far smaller. In opposed to any idea of atoms, and clashes with Boltz- both cases, though, there are clearly periods over mann were inevitable. Boltzmann hated them. In 1895 which the entropy decreases. All Johnson’s analysis is Mach became Professor of the History and Philos- extremely helpful to anybody wishing to understand ophy of Sciences at Vienna, where Boltzmann had Boltzmann’s achievements. become Professor of Theoretical Physics the year He also presents a series of vignettes of Boltzmann’s before. Boltzmann’s unease was such that he moved life — some amusing, such as his walking down the to Leipzig in 1900, returning to Vienna only in 1902, main street accompanied by a cow, purchased to after Mach’s retirement due to illness. provide his children with fresh milk, but more often Planck was (until his dramatic conversion) the last sad, including his disastrous manoeuvers attempting of the important exponents of classical thermody- to take up a chair in Berlin without informing his namics, following in particular Rudolf Clausius and current employer, the Austrian education minister, Hermann von Helmholtz. As such he detested Boltz- and a study of his self-destructive concern about mann’s statistical approach to matters of heat ow a student lecture to be given the following day, a and particularly the use of probability in physics. lecture he should have been able to give practically in Zermelo concentrated on the nature of entropy. A his sleep. He even provides a ‘happy ending’ in which crucial point in classical thermodynamics was that Boltzmann dies at peace, soothed by his daughter entropy never decreases. In certain very special cir- playing the Moonlight Sonata. Sadly, of course, this cumstances it might remain the same, but otherwise was not to be. it increases thus providing an arrow of time. On the Ironically, as Johnson points out, it was at precisely statistical de nition, on the other hand, the entropy this period that Boltzmann’s main ideas were being of very small systems would often decrease, but justi ed. It was argued, long before Einstein’s detailed as the size of the system increased the probability analysis in 1905, that Brownian motion, the irregular of entropy decreasing became smaller and smaller, motion of particles in a uid, was due to bombard- becoming in nitesimally small for systems of macro- ment by atoms; and more speci cally, Planck’s epoch- scopic size but never exactly zero. Zermelo plagued making study of black-body radiation in 1900, which Boltzmann on this point introduced the quantum theory, was based soundly on Boltzmann’s probabilistic arguments, which Planck It is well known that Boltzmann tragically hanged had long excoriated. In repentance, as Johnson men- himself in September 1906. The community of physi- tions, Planck twice proposed Boltzmann unsuccess- cists has traditionally absolved itself of blame for fully for the Nobel Prize for Physics. For Boltzmann this event, providing as evidence his various ill- this was all too late. This book is an interesting and nesses, both physical, including enormously dimin- thoughtful account of Boltzmann’s life and work. ished vision, asthma, chest pains and headaches, and mental, depression and in particular what Johnson Andrew Whitaker characterises as anxiety. Readers of Johnson’s slim book may judge for themselves to what extent physi- Andrew Whitaker is cists contributed to these illnesses. Johnson provides Emeritus Professor of detailed analysis of relatively small systems demon- Physics at Queen’s Uni- strating Boltzmann’s main achievements. He studies versity, Belfast, and the probabilities of students distributing themselves Chair of the Institute of Physics History of Physics group.

OBITUARIES 45 this did not stop him continuing to receive visitors, Obituaries of Members attend concerts and read widely. He never forgot his gratitude to University College School and Trinity Col- Ernst Sondheimer: – lege Cambridge, for their support given to a refugee from Nazi Germany in di cult times. Ernst Sondheimer, who Gerald Gould: – was elected a member of the London Mathemati- Gerald Gould, who was cal Society on 28 June elected a member of 1974, died on 9 June 2019, the London Mathemati- aged 95. cal Society on 18 March 1954, died on 23 Febru- Julian Sondheimer writes: ary 2019, aged 93. Ernst was born in 1923 in Germany. In 1937 the David Edmunds and Des family emigrated to London, and Ernst started his Evans write: Gerald was new life in England at University College School in born in Bermondsey, Hampstead. In 1942 he went up to Trinity College London, on 9 September 1925. From Wilson’s Gram- Cambridge and remained there as research student mar School he won an open scholarship to study and fellow until 1952. In 1946 he became a British physics at Christ’s College, Cambridge, an event citizen and in 1950 he married Janet Matthews, a his- celebrated by the school with a day o . After grad- tory fellow at Girton College, Cambridge. In 1951 he uating, he served his military service working for was appointed as a lecturer at Imperial College and Tube Alloys, the code name for the top security in 1955 he became a reader at Queen Mary College. research and development programme to develop nuclear weapons. That done, Gerald enrolled for a At the age of 37, in 1960, Ernst was appointed Pro- mathematics degree in Birkbeck College, University fessor of Mathematics at West eld College in the of London. There he came under the in uence of University of London, where he remained for 22 years Lionel Cooper, and after graduating started a PhD until his retirement in 1982. For most of this period under Cooper’s supervision. In 1950 Cooper was ap- he was Head of Department and he was instrumental pointed to the chair in what was then the University in creating a very successful mathematics research College of South Wales and Monmouthshire (now department. Ernst cared deeply about his students Cardi University), and Gerald went with him; he was and sta and is fondly remembered by them. to spend the rest of his career in Cardi . Apart from his mathematics, Gerald had many interests and tal- Earlier, as a young researcher in the eld of mathe- ents. He was an accomplished musician, being highly matical physics, his main achievement was a seminal regarded and sought after by orchestras as a tim- and groundbreaking paper in 1948 with G.E.H. (Harry) panist, bassoonist and autist. Bridge was another Reuter (also a German refugee) on the anomalous passion at which he excelled, winning a worldwide skin e ect. This was in the area of the optical prop- competition in 2006 involving 1,000 players on four erties of metals and involved the di cult solution continents. In 1954 Gerald developed TB and spent of an integro-di erential equation. Later Ernst co- time in an isolation hospital. He claimed that to authored the books Green’s Functions for Solid State contend with the boredom, he learned to smoke, Physicists and Numbers and In nity. became a master of Meccano, and learned Russian. He would continue to smoke with obvious pleasure Towards the end of his time at West eld, in 1981, until 1984, and his knowledge of Russian had a big Ernst and his family were hard hit by the death of part to play in the contribution he made to the his younger brother Franz, the eminent chemist (a mathematical community; he went on to act for the Fellow of the Royal Society, Franz was distinguished LMS as English Edition Editor of Sbornik, and trans- for his work on the synthesis of natural products). lated into English various Russian books, notably two After retiring from West eld College, Ernst revelled that appeared in the Encyclopaedia of Mathematical in his two passions of mountains and owers, taking Sciences series: Dynamical Systems by D.V. Anosov up the editorship of the Alpine Journal and devoting and General Topology III by A.V. Arhangel’skii. himself to his wonderful garden at his home in North London. At age 88, an unexpected di culty with a medical procedure left him in a frail condition, but

46 OBITUARIES Gerald’s early work evolved from his PhD on where he was awarded a personal chair and spent integration spaces, his principal contribution to the rest of his career. He was challenged by dif- research being perhaps his paper Integration Over ficult questions and, although he wrote only 17 Vector-Valued Measures which appeared in the Pro- papers, virtually all of them contain deep and ceedings of the LMS. This introduced a type of inte- clever ideas. He continued to work in geometric gral (the Gould integral) of a bounded real-valued measure theory, relating rectifiability of sets (i.e. function with respect to a finitely additive set func- whether they are ‘curve-like’ or ‘surface-like’) to tion taking values in a Banach space. Gerald had a local measure densities. But he had a broad inter- great appetite for new areas of mathematics and est across mathematical analysis. He constructed was able to absorb challenging new material with a clever counter-example to Khinchin’s conjecture enviable ease. Much of his research was gener- on the uniform distribution of certain sequences ated by problems and flaws discovered through his modulo 1. He was fascinated by variants of the extensive reading. His later work on test-function Kakeya set and curve packing problems and he spaces and various aspects of spectral theory are solved the problem of ‘the minimal comfortable liv- examples of this. ing quarters for a worm’ by showing that given any Outside his mathematics and his many interests, plane set of zero area (Lebesgue measure) there Gerald had a rich family life. He is survived by his are arbitrarily short smooth curves that cannot beloved wife Enid, his children Nina and Ben and be mapped into the set by a similarity (or, much three grandchildren. They will miss him sorely, as more generally, an analytic) transformation. Out- will his friends and colleagues. standing was his geometrical proof of the ‘circle packing conjecture’, of fundamental importance John Marstrand: – in harmonic analysis: given a plane set E of zero area, every circle with centre x intersects E in John Martin Marstrand, zero perimeter length, apart from a set of cen- who was elected a mem- tres x of zero area. In a probabilistic direction, ber of the London Math- John, together with Geoffrey Grimmett, answered ematical Society on 18 in 1990 the major outstanding question on site June 1953, died on 29 percolation on lattices in 3 or more dimensions. May 2019, aged 93. John was a highly regarded and engaging lecturer. Once, lecturing on ‘random walks’, he jumped onto Kenneth Falconer writes: the front bench, blindfolded himself and stepped Although he took his forwards and backwards at random until he even- DPhil at Oxford, John tually fell off the end of the bench! He was very was essentially supervised by Cambridge Professor careful in all his duties, often spending hours Abram Besicovitch. His thesis contained a number on tasks others would take a few minutes over. of remarkable results, notably ‘Marstrand’s projec- Once when examining a PhD thesis his attention tion theorems’ on the Hausdor dimensions of or- to detail resulted in a five and a half hour viva! thogonal projections of subsets of the plane onto Outside mathematics, John went through a range lines. Apart from generalisations by Pertti Mattila and of passionate interests, from fast sports cars to Robert Kaufman, this attracted little attention un- extreme skiing. In his 40s he took up fell-racing til the late 1970s when Mandelbrot popularised and and in 1982 became British over-50s fell-running uni ed the notion of fractals. It was then realised champion. He regularly participated in the Ben that John’s work was just what was needed for the Nevis Race and on several occasions won the over- study of the geometry and dimensions of fractals, 40s or over-50s sections, with his fastest time in and his projection theorems became the prototype 1983 an incredible 1 hour 48 minutes 54 seconds. for the now ourishing area of fractal geometry. In- John was a kind and considerate person and is deed, the paper based on John’s thesis that appeared remembered by his former colleagues as a ‘love- in the LMS Proceedings in 1954 was highlighted in able eccentric’. He was modest about his achieve- 2015 as one of six ‘landmark’ papers published in ments but his determination and insight when he the Proceedings during its rst 150 years. became absorbed by a project led to some quite remarkable work of enduring quality. After short periods in Cambridge and Sheffield, John moved to the University of Bristol in 1958

OBITUARIES 47 Christopher Hooley: – to these ideas. In particular the use of exponential sums, as pioneered by Hooley, is now ubiquitous. Christopher Hooley, who Hooley’s writing was well known for his love of such was elected a member of arcane phrases as ‘the dexter side of the ante- the London Mathemati- penultimate equation’. Equally however his papers cal Society on 28 June were a model of clarity and accuracy. They were often 1974, died on 13 Decem- couched in seemingly old-fashioned terms, both lin- ber 2018, aged 90. guistically and mathematically, but they never failed to enlighten. Roger Heath-Brown and Christopher Hooley had many interests, most of Thomas and Adam which were pursued with vigour. From early on as Hooley write: Professor a pupil at Abbotsholme School during the Second Christopher Hooley, FRS, FLSW was one of the lead- World War he developed a love not only of mathemat- ing analytic number theorists of his day. He won ics, but also of classical, military, and naval history. In Cambridge University’s Adams Prize in 1973, and the years to come he would delight in discussing the his- LMS Senior Berwick Prize in 1980. He was elected tory of western and eastern Roman empires with his a Fellow of the Royal Society in 1983, and was a colleagues from the history department at Cardi , Founding Fellow of the Learned Society of Wales. He who would ruefully admit they were out of their com- gave a plenary lecture at the 1983 ICM in Warsaw. fort zone, using that standard excuse ’not my period’! — Hooley counted all European history of the two Christopher Hooley did his undergraduate degree at millennia A.D. as his period. This interest went along Corpus Christi College, Cambridge, and continued with a love of antiques and especially of collecting there with a PhD under Albert Ingham. His career West Country friendly society brasses. Many days then took him to Bristol, to Durham, and nally to were spent scouring antique shops and following up Cardi , where he served as Head of the School of leads to remote places in Somerset during the 1970s Mathematics from 1988 to 1995. and 80s. His wife, Birgitta, denied pets as a child in Sweden, made sure that the family always had plenty He was the author of nearly 100 papers. The best of dogs — at one time six, sometimes cats as well: known of these concerns Artin’s conjecture on primi- this interest led to another hobby, that of travelling tive roots. The conjecture proposes that primes with to terrier shows in the summer months, from which a given primitive root have a de nite density, and Birgitta, Christopher, and their two sons, Thomas gives a formula for what the density should be. In and Adam, often returned proudly bearing coloured particular it would follow that 10, for example, is a rosettes for their dogs’ successes. Supportive as a primitive root for in nitely many primes p, so that the father he was nonetheless quick to spot and point decimal expansion for 1/p would recur with period out inaccuracies of memory in the reminiscences of p − 1. Hooley’s work, in 1967, proved the conjecture his sons at his 90th birthday party, held in Bristol in full, assuming the Riemann Hypothesis for a class just four months before he died in December 2018. of Dedekind zeta-functions. These would have been Christopher’s grandfather had taught carpentry at highly abstruse objects to analytic number theorists Maccles eld and on inheriting his tools Christo- in those days, and even now it seems remarkable pher set about using them to restore or completely that they could have any bearing on such a concrete remake a large number of period sash windows in problem about decimal fractions. the family home in Somerset, often at some con- siderable height. Once done, he turned his hand to Many of Hooley’s papers made use of estimates for panelling, lath and plaster, and similar skills with great exponential sums originating in the work of Weil and success. Deligne. Thus for example, he was able to establish The practical side was also present when it came to the Hasse Principle for non singular integral cubic motoring. The rst family car, a 1950s’ Wolseley 6/80, forms in 9 or more variables, by using information was kept and later given a thorough restoration in about the associated exponential sums. Another the 1980s. Christopher showed no fear in creating recurrent theme was the introduction of novel sieve an entire wiring loom from scratch! methods, and his Cambridge Tract “Applications of sieve methods to the theory of numbers” showcases the impressive variety of sieve tools he developed. Modern analytic number theory owes a great deal

48 OBITUARIES These interests and his family life provided the coun- publications at that time was his book on Degree terpoint to Hooley’s academic career. At times uncon- Theory in the Cambridge Tracts in Mathematics se- ventional, for instance researching deep into the night ries; this book was particularly well received and with a beloved whisky and cigarette alongside an became an established text and was widely con- eclectic mixture of steam railway magazines, Dante, sulted. During the 1980s his interests moved towards Milton, and paperback thrillers, he had a ready wit the study of the second part of Hilbert’s Sixteenth and excellent sense of humour. He will be remem- Problem on the number and relative con guration bered with fondness by colleagues and family alike. of limit cycles of polynomial systems in the plane. This is acknowledged to be a very di cult area of Noel G. Lloyd: – research but it is internationally recognised that he and his research group have been responsible for Professor Noel Lloyd, substantial developments, and his loss is strongly who was elected a mem- felt amongst researchers in this area. Their progress ber of the London Math- involved innovative use of computer algebra, as well ematical Society on 15 as analytic approaches. February 1973, died on 7 In the 1980s Noel was a member of the Mathematics June 2019, aged 72. Committee of the SERC where his detailed knowl- edge and his meticulous fairness came to the fore. Alun Morris writes: Noel Also, in 1986–90 he was Joint Editor-in-Chief with sadly died after a pro- me of the Journal of the LMS — a dramatic period longed ght against for the journal with an overnight need to move the prostate cancer which he met with his usual quiet printing to the Cambridge University Press. He was composure. This followed a full life distinguishing also a long-serving Editor of the Mathematical Pro- himself both as a mathematician and in his later ceedings of the Cambridge Philosophical Society. He years as a senior university administrator and in was promoted to Senior Lecturer, Reader and then public life. in 1986 to a Personal Chair. Although at that point not heavily involved in administration, he was highly Noel was born in Llanelli, Carmarthenshire. After respected and was for a short time Head of the attending Llanelli Grammar School, where his father Department of Mathematics, Dean of the Faculty of was a mathematics teacher, he entered Queens’ Col- Science and then Pro Vice-Chancellor, then Registrar lege, Cambridge in 1965 having obtained an Entrance and Secretary of the University and nally in 2004 Scholarship. He graduated with a B.A. in 1968 with its Vice-Chancellor. He held that position with great a rst class in each year and an MA in 1972. On distinction until 2011. In 2010 he was appointed CBE graduating, he worked for his PhD in a research area for Services to Higher Education in Wales and in 2011 which was not too fashionable at the time, ordinary a Fellow of the Learned Society of Wales. di erential equations, under the supervision of Sir The many tributes that have appeared after his Peter Swinnerton-Dyer. His PhD thesis dealt with a untimely death are an indication of the esteem he case of van der Pol’s equation not considered by was held as Vice-Chancellor. He was meticulous in J.E. Littlewood and Mary Cartwright. In addition to his preparation, quietly firm and fair. He was univer- numerous college prizes he also received the pres- sally regarded as a generous and compassionate man tigious Rayleigh Prize. He proceeded to a Research and a person of great integrity, respected equally by Fellowship at St John’s College, Cambridge. After two his academic colleagues and fellow administrators. years in that position he was enticed to a Lectureship During his tenure as Vice-Chancellor he was Chair in Pure Mathematics at what was then the University of Higher Education Wales and Vice-President of College of Wales, Aberystwyth (now Aberystwyth Uni- Universities UK. He also served on many other UK versity), which he took up on 1 January 1975. At that university bodies. On his retirement, his talents were time, it was not anticipated that the university was in great demand. He became an independent mem- appointing its future Vice-Chancellor — certainly not ber of the Silk Commission established by the UK by him. He did not need to be persuaded; he had government to look at the future of devolution in long expressed his wish to come to his father’s alma Wales. He also served as a lay member of the Judicial mater. His research work ourished in his chosen Appointments Commission. For six years he was area of applied analysis or non-linear analysis. He Chair of Fair Trade Wales. established a strong school in that area supervising a number of good PhD students. Among numerous

OBITUARIES 49 Throughout his life, his church was central to his In 1998, Stuart again changed direction and built life. He served not only his church with distinction on his research experience to found and edit a but also his denomination and other religious causes BMJ publication Clinical Evidence, which assessed both locally and nationally. Above all, he was a bril- the clinical significance of medical treatments and liant organist; as one tribute said, students would drugs. In 2002 Stuart returned to GP practice in see their VC relaxing on a Sunday morning at the London. With Dr Elizabeth Robinson, he took over a organ playing his favourite Bach. Noel married Dilys failed GP practice in the Kings Cross area and they in 1970; next year would have been their golden wed- worked together there for 8 years providing excel- ding. They had a son Hywel and daughter Carys and lent health care to some of the most disadvantaged two grand-daughters Sioned and Catrin. communities in London. Stuart retired from GP practice in 2010, in order to Stuart Brian Barton: – devote much of his time to mathematics, his first academic love. He began to study with the Open Dr Stuart Barton MD, University, completing a first class degree in mathe- DPhil, who was elected a matics in 2017. His tutors speak highly of his assign- member of the London ment work, which was always of excellent quality Mathematical Society and often contained material that went well beyond on 10 November 2017, what the assignments officially required. After mov- passed away in hospital ing to live in Cambridge, he regularly attended the in Cambridge on 12 April Open University’s maths tutorials there and also 2019, aged 66. weekend schools, where he greatly enjoyed meeting other people with similar interests, and is remem- Phil Rippon writes: As a bered by tutors as one of the brightest and liveliest young person at Oldham Hulme Grammar School, students. His partner Elizabeth Robinson reports Stuart excelled at mathematics and physics, but that he read widely about the subject, and was was advised that medicine was a more appropriate especially influenced by the life of Paul Erdo´´s, as career. Accordingly, he read medicine at Balliol Col- described in the book The Man Who Loved Only lege, Oxford, and took a DPhil there in physiology Numbers by Paul Hoffman. Like many mathemati- in 1977, working much of the time in Stony Brook, cians Stuart would enjoy long walks (with their dog New York with Professor Ira Cohen. They had been Mabel), while thinking about mathematics problems. friends at Oxford and in fact published together a Stuart then embarked on an MSc in Mathematics rather mathematical paper on transmitter release at the Open University, his fascination with number from nerve terminals (appearing in Nature in 1977). theory (and especially Goldbach’s conjecture) lead- Stuart then began a career as a GP in the UK and ing him to study modules based on Tom Apostol’s also quali ed as a consultant physician though he book on analytic number theory. It was at this time chose to remain a GP. that he joined the LMS, confessing to me (as his LMS proposer) that his dream was to add publications in In 1993, Stuart changed direction and became a pure mathematics to his publications in medicine lecturer at the University of Liverpool, specialis- and biology. His tutors say that he would surely have ing in evidence based medicine, where he could achieved this dream in time, but sadly it could not bring into play his love of mathematics (espe- happen. In April 2019, he suffered a brain aneurysm cially statistics). He became a leader in that field, and died in hospital some days later. He is survived publishing many articles, with co-authors such as by his partner Dr Elizabeth Robinson, and by his Professor Tom Walley CBE. two adult children Jonathan and Andrew.

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